REGAININGLY APPROXIMABLE NUMBERS AND SETS

. We call an α ∈ R regainingly approximable if there exists a computable nondecreasing sequence ( a n ) n of rational numbers converging to α with α − a n < 2 − n for inﬁnitely many n ∈ N . We also call a set A ⊆ N regainingly approximable if it is c.e. and the strongly left-computable number 2 − A is regainingly approxi-mable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly appro-ximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K - trivial, we construct such an α such that K ( α ↾ n ) > n for inﬁn-itely many n . Similarly, there exist regainingly approximable sets whose initial segment complexity inﬁnitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.


Introduction
We call a sequence (a n ) n of real numbers increasing if, for all n ∈ N, a n < a n+1 , and nondecreasing if, for all n ∈ N, a n ≤ a n+1 , and we define the terms decreasing and nonincreasing analoguosly.A real number is called left-computable if there exists a computable nondecreasing sequence of rational numbers converging to it; note that some textbooks [11,4] call such numbers left-c.e..A real number α is called E-mail address: peter.hertling@unibw.de,r@hoelzl.fr,philip.janicki@unibw.de.
computable if there exists a computable sequence (a n ) n of rational numbers satisfying |α − a n | < 2 −n , for all n ∈ N. It is easy to see that any computable real number is left-computable.In this article, we study real numbers that are limits of computable, nondecreasing, converging sequences (a n ) n of rational numbers which are not required to satisfy the condition |α − a n | < 2 −n for all but only for infinitely many n ∈ N. Definition 1.We call a real number α regainingly approximable if there exists a computable nondecreasing sequence of rational numbers (a n ) n converging to α such that α − a n < 2 −n holds for infinitely many n ∈ N.
Intuitively speaking, regainingly approximable numbers are not required to possess approximations that converge as speedily as those to computable numbers, but they must possess approximations that "catch up" with that speed infinitely often while being allowed to "dawdle" arbitrarily in between.Trivially, every regainingly approximable number is left-computable, and every computable number is regainingly approximable.In fact, even the following stronger observation holds.
Example 2. Every non-high left-computable number is regainingly approximable.Indeed, let α be a left-computable number that is not high.Fix a computable nondecreasing sequence of rational numbers (a n ) n converging to α.An increasing function s : N → N with α − a s(n) < 2 −n for all n ∈ N can be computed from oracle α.Since α is not high, according to a characterization of highness by Martin [8] (see Soare [15,Theorem XI.1.3]),there exists a computable, w.l.o.g.increasing function r : N → N with r(n) ≥ s(n) for infinitely many n ∈ N. Then for any n ∈ N with r(n) ≥ s(n) we obtain Therefore, α is regainingly approximable.
The remainder of the article is structured as follows: • In Section 3 we begin by showing that Definition 1 is robust under minor modifications, with the equivalences between the different formulations turning out to be effectively uniform.We also state some computability-theoretic properties of the set {n ∈ N : α − a n < 2 −n }, that is, the set of points where a regaining approximation (a n ) n "catches up." • We begin Section 4 by giving several characterizations of the sets A ⊆ N that have the property that the real number is regainingly approximable.For the rest of the section we then focus on c.e. sets A ⊆ N: A real number x ∈ [0, 1] is called strongly left-computable if there exists a computably enumerable set A ⊆ N with x = 2 −A .It is well-known that the set of strongly left-computable numbers is a proper superset of the set of computable numbers in [0, 1] and a proper subset of the set of left-computable numbers in [0, 1].We consider c.e. sets A such that 2 −A is regainingly approximable and call such sets regainingly approximable.We give different characterizations of this class of sets, and note that, unlike for regainingly approximable numbers, not all arguments here can be fully effectively uniform.• In Section 5, we state further computability-theoretic properties of regainingly approximable sets.First, we observe an easy splitting result, namely that every c.e. set C ⊆ N is the union of two disjoint regainingly approximable sets A, B ⊆ N. Next, we prove that there is a c.e. set that is not regainingly approximable.Finally, we show that every c.e. Turing degree contains a regainingly approximable set.• In Sections 6 and 7 we look at the Kolmogorov complexity of regainingly approximable numbers and sets.On the one hand, we observe that every regainingly approximable number is i.o.Ktrivial and, hence, not Martin-Löf random.On the other hand, we show that there exists a regainingly approximable number such that the prefix-free Kolmogorov complexity of infinitely many of its initial segments (in a sense to be explained in Section 2) exceeds their length, and that there exist regainingly approximable sets whose initial segments infinitely often have the maximal Kolmogorov complexity that is possible for a c.e. set.• In Section 8 we observe that regainingly approximable sets and numbers behave badly with respect to arithmetical operations: The set of regainingly approximable sets is closed neither under union nor under intersection, and while the set of regainingly approximable numbers is closed downwards under Solovay reducibility it is not closed under addition.
• Finally, in Section 9 we formulate a uniformly effective algorithm for splitting c.e. sets into two regainingly approximable sets, which strengthens the splitting result of Section 5.In fact, this algorithm proves a stronger result, namely that every real that is regular in the sense of Wu [18] can be split into two regular reals that are additionally regainingly approximable.We close this introduction by mentioning that isolating the notion of regaining approximability was a key step needed for Hölzl and Janicki's [6] negative answer to the question posed by Merkle and Titov [10] whether among the left-computable numbers, being Martin-Löf random is equivalent to being non-speedable.

Notation
For general background on computability theory and algorithmic randomness, we refer the reader to the usual textbooks [15,11,4].We will use Cantor's pairing function •, • : for all m, n ∈ N.This is a computable bijection, so let π 1 : N → N and π 2 : N → N denote the two components of its inverse function, that is, For n ∈ N, we write log(n) for the length of the binary representation of n, thus log(n) = ⌊log 2 (n)⌋ + 1 if n > 0 and log 2 (0) = 1.Hence, log(n) is identical to the usual binary logarithm up to a small additive constant.
For functions f, g : N → N we write f ≤ + g if there exists a constant c ∈ N such that f (n) ≤ g(n) + c for all n.A set A ⊆ N will often be identified with its characteristic sequence, that is, we define We end this section by laying out the following conventions about how, in the remainder of the article, we will (often tacitly) identify real numbers with infinite binary sequences and subsets of the natural numbers: • If we ascribe to some x ∈ R a property that is formally defined only for infinite binary sequences, then we mean that a specific, uniquely determined infinite binary sequence has that property; namely, the sequence containing infinitely many ones and that is a binary representation of the unique real number y ∈ (0, 1] with x − y ∈ Z. • If we ascribe to some x ∈ R a property that is formally defined only for subsets of N, then we mean that the uniquely determined infinite set A ⊆ N, whose characteristic sequence is the infinite binary sequence defined in the previous item, has that property.• When we talk about "initial segments" of some x ∈ R, then we are again referring to the initial segments of the infinite binary sequence defined in the first item.This will be used for instance when we talk about the plain or prefix-free Kolmogorov complexity of initial segments of some such x.• Similarly, when we talk about the initial segments of an A ⊆ N, then we are referring to its characteristic sequence.

Robustness
In this section, we first show that slight changes to the definition of regainingly approximable numbers do not lead to a different notion.The following lemma will be useful; note that no computability assumptions are made.
Lemma 3. Let (a n ) n be a nondecreasing sequence of real numbers converging to some real number α such that, for infinitely many n ∈ N, α − a n < 2 −n .Then, for every unbounded function f : Proof.By assumption, the set for n ∈ A. The function g is well-defined because f is unbounded.For every n ∈ A we have f (g(n)) ≤ n < f (g(n) + 1).The set There are several obvious modifications of Definition 1 that one might want to consider.First, instead of allowing nondecreasing (a n ) n , we might require (a n ) n to be increasing.Secondly, one might replace the condition α − a n < 2 −n by the condition α − a n < 2 −f (n) where f : N → N is an arbitrary computable, unbounded function of one's choice; or, one might ask for this to hold only for some fixed computable, nondecreasing, unbounded function f : N → N, a seemingly weaker requirement.However, it turns out that none of these modifications make any difference.
Proposition 4. For a real number α ∈ R the following statements are equivalent: (1) α is regainingly approximable.
(2) There exists a computable, increasing sequence of rational numbers (a n ) n converging to α such that, for infinitely many n ∈ N, α − a n < 2 −n .(3) For every computable, unbounded function f : N → N there exists a computable increasing sequence of rational numbers (a n ) n converging to α such that, for infinitely many n ∈ N, (4) There exist a computable, nondecreasing, and unbounded function f : N → N and a computable nondecreasing sequence of rational numbers (a n ) n converging to α such that, for infinitely Note that this implies in particular that it makes no difference whether we use "<" or "≤" in the definition of regaining approximability.We would also like to point out that all implications in the following proof are uniformly effective. Proof.
(1) ⇒ (3): Let α be a regainingly approximable number.Let (b n ) n be a computable nondecreasing sequence of rational numbers converging to α with α − b n < 2 −n for infinitely many n ∈ N. Let f : N → N be a computable, unbounded function.Then the function g : N → N defined by g(n) := 1 + n + max{f (m) : m ≤ n} is computable, increasing, and satisfies g(n) ≥ f (n) + 1, for all n ∈ N. In particular, g is unbounded.The sequence (a n ) n of rational numbers defined by is computable and increasing and converges to α.By Lemma 3 there exist infinitely many n with α − b g(n+1) < 2 −g (n) .For all such n we obtain (1) ⇒ (4): Trivial.(4) ⇒ (1): Assume that f : N → N is a computable, nondecreasing, and unbounded function and (b n ) n is a computable nondecreasing sequence of rational numbers converging to α such that, for infinitely many n ∈ N, α − b n < 2 −f (n) .Define a function g : N → N via g(0) := max{m ∈ N : f (m) = f (0)} and g(n + 1) := max{m ∈ N : f (m) = f (g(n) + 1)}, for n ∈ N; informally speaking, the graph of f is an infinite increasing sequence of plateaus of finite length and g(n) gives the x-coordinate of the right-most point in the (n + 1)-th plateau.Clearly, g is computable and increasing and satisfies f (g(n)) ≥ n for all n ∈ N. Furthermore, for every k ∈ N there exists exactly one n ∈ N with f (k) = f (g(n)), and this n satisfies k ≤ g(n).The sequence (a n ) n of rational numbers defined by a n := b g(n) , for all n ∈ N, is computable and nondecreasing and converges to α.By assumption, the set is infinite as well.Consider a number n ∈ A, and let k ∈ B be a number with Next, we observe that if a left-computable number α is regainingly approximable then this will be apparent no matter which of its effective approximations we look at.Proposition 5. Let α be a left-computable number, and let (a n ) n be a computable, nondecreasing sequence of rational numbers converging to α.Then the following conditions are equivalent.
(2) There exists a computable, increasing function r : N → N such that, for infinitely many n, α − a r(n) < 2 −n .
Note that the proof is effectively uniform in both directions. Proof.
(2) ⇒ (1): Assume that there exists a computable, increasing function r : N → N such that we have α − a r(n) < 2 −n for infinitely many n.Then the sequence (b n ) n of rational numbers defined by b n := a r(n) is computable, nondecreasing, converges to α, and satisfies, for infinitely many n, α − b n < 2 −n .Hence, α is regainingly approximable.
(1) ⇒ (2): Assume that α is regainingly approximable.By Proposition 4 there exists a computable, increasing sequence (b n ) n of rational numbers converging to α such that there exist infinitely many n with α − b n < 2 −n .We define a computable, increasing function r : N → N by r(0) := min{m ∈ N : a m ≥ b 0 }, and for n ∈ N.For all n ∈ N we have a r(n) ≥ b n .Thus, for the infinitely many n ∈ N with α − b n < 2 −n , we obtain We close this section by investigating the computability-theoretic properties of the set of points where a regaining approximation "catches up."Proposition 6.Let α be a regainingly approximable number, and let (a n ) n be a computable nondecreasing sequence of rational numbers converging to α with α − a n < 2 −n for infinitely many n ∈ N. Then has the following properties: (1) N \ A is computably enumerable.
(2) The following are equivalent: Clearly, f computably enumerates N \ A. Suppose that A is not hyperimmune.Then there exists a computable function r : for all n ∈ N. Hence, α is computable.

Regainingly Approximable Sets
In this section we study sets A ⊆ N such that 2 −A is regainingly approximable.We start by characterizing these sets in ways analogous to Propositions 4 and 5, before then focusing on the particularly interesting case where A is c.e.We need the following terminology and easy lemma: Following Soare [15] we call a sequence (A n ) n of finite sets A n ⊆ N a strong array if the function n → i∈An 2 i is computable, and a strong array (A n ) n a uniformly computable approximation from the left of a set for all i ∈ N, lim n→∞ A n (i) = A(i).
Lemma 7 (Soare [14]).For a set A ⊆ N the following conditions are equivalent.
(2) There exists a uniformly computable approximation from the left of the set A ⊆ N.
With this we are ready to establish the following characterization.
Theorem 8. Let A ⊆ N be a set such that the number 2 −A is leftcomputable.Then the following conditions are equivalent.
(2) There exists a uniformly computable approximation (A n ) n of A from the left such that, for infinitely many n, A ∩ {0, . . ., n − 1} = A n .
(3) For every uniformly computable approximation (B n ) n of A from the left there exists a computable, increasing function s : N → N such that, for infinitely many n, Proof.
(3) ⇒ (2): By Lemma 7 some uniformly computable approximation (B n ) n of A from the left exists.Then by assumption a function s as in (3) must exist as well.Thus, (2) follows immediately by letting (2) ⇒ (1): Let (A n ) n be a uniformly computable approximation of A from the left such that A ∩ {0, . . ., n − 1} = A n for infinitely many n.
The sequence (2 −An ) n is a nondecreasing computable sequence of rational numbers converging to 2 −A .For the infinitely many n with (1) ⇒ (3): If A is cofinite then it is easy to see that for any computable approximation of A from the left there is a function s as required by (3); thus w.l.o.g.let A be coinfinite.Let (B n ) n be an arbitrary uniformly computable approximation of A from the left.Then (2 −Bn ) n is a computable, nondecreasing sequence of rational numbers converging to 2 −A .By Proposition 5 there exists a computable increasing function r : N → N such that, for infinitely many n, 2 −A − 2 −B r(n) < 2 −n .If there are infinitely many n with and note that this is well-defined as lim n→∞ B n (i) = A(i) for all i ∈ N. It is clear that s is increasing and computable.Fix any of the infinitely many n ≥ N with 2 For the sake of a contradiction, assume otherwise; then one of (iii For the remainder of this section we will focus on c.e. sets.
Definition 9. We call a set A ⊆ N regainingly approximable if it is computably enumerable and the real number 2 −A is regainingly approximable.
That we resort to the number 2 −A in this definition concerning sets may seem unnatural and raises the question whether regainingly approximable sets could alternatively be defined using enumerations of their elements.This is indeed possible, as we will show now.
We call a total function f : If f (k) = n+1 then we say that f enumerates n into A at stage k.Note that here f (k) = 0 encodes that the function f does not enumerate anything into A at stage k.It is clear that a set A ⊆ N is computably enumerable if and only if there exists a computable enumeration of A.
Definition 10.Let r : N → N be a nondecreasing, unbounded function.We call an enumeration f : N → N of a set A ⊆ N r-good if there exist infinitely many n such that Remark 11.We call an enumeration of some A an enumeration without repetitions if for every n ∈ A there exists exactly one k ∈ N with f (k) = n + 1.From a given enumeration f of A one can easily compute one without repetitions by defining, for all k ∈ N, Note that if f was r-good for some nondecreasing, unbounded function r then f is r-good as well.
Example 12. Let A ⊆ N be a decidable set.Then the function is a computable and id N -good enumeration without repetitions of A.
Theorem 13.For a c.e.A ⊆ N the following conditions are equivalent.
(2) There exists a computable id N -good enumeration of A.
(3) There exists a computable, nondecreasing, unbounded function r : N → N such that there exists a computable r-good enumeration of A. (4) For every computable enumeration f of A there exists a computable, increasing function r : N → N such that f is r-good.
In particular, in analogy to Proposition 5, if a set A is regainingly approximable then this will be apparent no matter which of its effective enumerations we look at.
Proof of Theorem 13. (3) ⇒ (1): Let r : N → N be a computable, nondecreasing, unbounded function, and let f : N → N be a computable r-good enumeration of A. Then (a n ) n defined for all n ∈ N via is a computable, nondecreasing sequence of rational numbers converging to 2 −A ; and since we have for infinitely many n that (1) ⇒ (4): Let A ⊆ N be a c.e. set such that 2 −A is regainingly approximable, and let f : N → N be an arbitrary computable enumeration of A. Then (a n ) n defined by a n := 2 −Enum(f ) [n] , for all n ∈ N, is a computable nondecreasing sequence of rational numbers converging to 2 −A .By Proposition 5 there exists a computable, increasing function r : N → N such that, for infinitely many n, We obtain {0, . . ., n − 1} ∩ A ⊆ Enum(f )[r(n)] for infinitely many n.Hence, f is r-good.(4) ⇒ (3): Trivial, as every c.e. set has a computable enumeration.
(3) ⇒ (2): Assume that r : N → N is a computable, nondecreasing, unbounded function and that f : . ., M n forms a disjoint partition of L n for every n.We let g : N → N be the enumeration that first enumerates all elements of M 1 in increasing order, then those of M 2 in increasing order, and so on.More formally speaking, g : N → N is defined as follows: For every n ∈ N, let m n be the cardinality of M n , let k mn−1 be its elements in increasing order and, for t with 0 ≤ t ≤ m n − 1, define This is well-defined due to A's infinity.Clearly, g is a computable enumeration (in fact, without repetitions) of A with g(n) = 0 for all n.We claim that it is in fact an id N -good enumeration of A. To see this, fix any of the, by assumption, infinitely many n with By construction, g enumerates exactly the elements of during the first j<n m j stages and, by choice of n, all elements of {0, . . ., n − 1} ∩ A that are not enumerated during these stages must be in M n .In fact, they are exactly all elements of M n , and thus will be enumerated by g in the immediately following stages, starting with stage j<n m j .As there cannot be more than n − j<n m j such numbers, this process will be completed before stage n.Remark 14.Note that "(3) ⇒ (2)" is the only implication in Theorem 13 whose proof is not fully uniformly effective; in its proof we non-uniformly distinguished whether A is finite or not.A simple topological argument shows that this is unavoidable; in fact, there does not even exist a continuous function F : N N → N N mapping every 2id Ngood enumeration without repetitions of an arbitrary A ⊆ N to an id N -good enumeration of A. To see this, we use the following two facts: (i) There exists only one enumeration of the empty set, namely the constant zero function 0 : N → N, which is an id N -good enumeration.(ii) For every function f : N → N that is an id N -good enumeration of N we have f (n) = 0 for all n ∈ N. Now assume, for the sake of a contradiction, that there is an F as described.By (i), we must have F (0) = 0, and as F was assumed to be continuous, there exists an n 0 ∈ N with F (0 n 0 N N ) ⊆ 0N N ; that is, F must map all inputs starting with sufficiently many 0's to a sequence starting with at least one 0. Define g : N → N as an enumeration of N "delayed by n 0 stages", that is, let for n ∈ N. Then while g is a 2id N -good enumeration without repetitions of N, we have F (g) ∈ 0N N , and no such enumeration can be an id N -good enumeration of N, contradiction.

Computability-Theoretic Properties
We begin this section by observing an easy splitting theorem that we will need later.Next, we will show that there are c.e. sets that are not regainingly approximable.Finally, we will show that every c.e. Turing degree contains a regainingly approximable set.The theorem leaves open whether the splitting can be done effectively.The answer is yes: in Section 9 we will present a uniformly effective algorithm that even works for a larger class of objects than just the c.e. sets.As corollaries to Theorem 15 we obtain two easy ways to see that the regainingly approximable sets are a strict superclass of the decidable ones.
Corollary 16.There exists a regainingly approximable set A ⊆ N that is not decidable.First proof.By Example 2, any c.e. set that is neither decidable nor high is regainingly approximable.
Second proof.Let C ⊆ N be an undecidable c.e. set.By Theorem 15 there are two disjoint regainingly approximable sets A, B such that C = A ∪ B. At least one of A or B must be undecidable.
Next, we separate regaining approximability from computable enumerability.
Theorem 17.There exists a c.e. set A ⊆ N that is not regainingly approximable.
Proof.Let ϕ 0 , ϕ 1 , ϕ 2 , . . .be a standard enumeration of all possibly partial computable functions with domain and range in N. As usual, we write ϕ e (n)[t]↓ to express that the e-th Turing machine (which computes ϕ e ) stops after at most t steps on input n.We shall construct a computable enumeration g : N → N of a set A ⊆ N such that the following requirements (R e ) will be satisfied for all e ∈ N: (R e ) : if ϕ e is total and increasing then ( According to Theorem 13 this is sufficient. We construct g in stages; in stage t we proceed as follows: Define e := π 1 (π 1 (t)) and k := π 2 (π 1 (t)), hence, e, k = π 1 (t).Check whether the following conditions are satisfied: (∀n ≤ e, k + 1 ) ϕ e (n)[t]↓ and (∀n < e, k + 1 ) ϕ e (n) < ϕ e (n + 1) and t ≥ ϕ e ( e, k + 1 ) and e, k ∈ Enum(g)[t].
If they are, set g(t) := 1 + e, k , otherwise g(t) := 0. This completes the construction; we proceed with the verification.It is clear that g is computable and an enumeration without repetitions of some c.e. set A ⊆ N. We wish to show that R e is satisfied for all e ∈ N. Consider an e ∈ N such that ϕ e is total and increasing, as well as a number n > e, 0 .There exists a unique k ∈ N with e, k < n ≤ e, k + 1 .The function g enumerates e, k into A at some uniquely determined stage t, that is, there exists exactly one t ∈ N with g(t) = 1 + e, k .Then Since n ≤ e, k + 1 , we have ϕ e (n) ≤ ϕ e ( e, k + 1 ) ≤ t, and therefore Thus e, k ∈ {0, . . ., n − 1} ∩ A witnesses that R e is satisfied with n e = e, 0 .
Finally, we show that the regainingly approximable Turing degrees are exactly the c.e. Turing degrees.
Theorem 18.For every computably enumerable set A ⊆ N there exists a regainingly approximable set B ⊆ N with A ≡ T B.
Proof.W.l.o.g.we may assume that A is infinite.Thus fix some computable injective function f : N → N with f (N) = A.
We will build a c.e. B by defining a computable injective function g : N → N and letting B := g(N).In parallel, we will define an increasing sequence (S i ) i such that (i) for all i we have {0, . . ., S i −1}∩B ⊆ {g(0), . . ., g(S i −1)}, which implies that B is regainingly approximable, and such that (ii) To ensure these properties, we will define (S i ) i in such a way that, for all i and for all t ≥ S i , we have on the one hand that f (t) ≥ i and on the other hand that g(t) ≥ S i .This last property will imply (i); and concerning (ii), informally speaking, our choice of (S i ) i means that f and g can enumerate "small" numbers only for arguments smaller than S i .This will allow us to show A ≤ T (S i ) i and B ≤ T (S i ) i .Finally, the statements (S i ) i ≤ T A and (S i ) i ≤ T B will follow from the way we define (S i ) i alongside g.
After these informal remarks, we proceed with the full proof.The following algorithm works recursively in infinitely many stages to compute two functions g : N → N and s : N 2 → N; we write s i [t] for s(i, t).Since it will turn out below that (s i [t]) t is eventually constant for every i, allowing us to define S i := lim t→∞ s i [t], it is suggestive to think of s i [t] as our preliminary guess for S i at stage t.At stage 0 we let s i [0] := i for all i ∈ N.And for every t ∈ N, at stage t + 1 we define and set for all i ∈ N. Finally, we define B := g(N).This ends the construction, and we proceed with the verification.First note that B is clearly computably enumerable.Claim 1.For every t ∈ N, the sequence (s i [t]) i is increasing.Proof.This is clear for t = 0, and can easily be seen to hold for all other t by induction.♦ Claim 2. For every i ∈ N, the sequence (s i [t]) t is nondecreasing and eventually constant.Proof.By construction (s i [t]) t is nondecreasing for every i ∈ N. Fix some i ∈ N; to see that (s i [t]) t is eventually constant, choose t i such that f (t) ≥ i for all t ≥ t i .Note that such a t i exists because (f (t)) t tends to infinity.Then, for every t ≥ t i , we have ♦ Claim 4. For every i ∈ N and every t ≥ S i we have f (t) ≥ i. Proof.Consider any i ∈ N and some stage t ≥ S i .If we had f (t) < i, then we would obtain ♦ Claim 5.For every i ∈ N and every t ≥ S i we have g(t) ≥ S i .Proof.Consider any i ∈ N and some stage t ≥ S i .By Claim 4, f (t) ≥ i, and by Claim 1, (s j [t]) j is increasing.Together with Claim 3, we obtain ♦ Claim 6. B is regainingly approximable.Proof.According to Claim 5 the function n → g(n) + 1 is an id N -good, computable enumeration of B, and hence B is regainingly approximable.♦ Claim 7. The function g is injective.
• If f (t 1 ) < f (t 2 ), then due to Claims 1 and 2 we have ] due to Claim 2 and we obtain Thus, g is injective.♦ Claim 8a.A is computable in (S i ) i .Proof: Given oracle (S i ) i , in order to decide whether n ∈ A for arbitrary n ∈ N, it suffices to check whether n is enumerated by f before stage S n+1 .If not, then since by Claim 4 we have f (t) ≥ n + 1 for all t ≥ S n+1 , we must have n / ∈ A. ♦ Claim 8b.B is computable in (S i ) i .Proof: Given oracle (S i ) i , in order to decide whether n ∈ B for arbitrary n ∈ N we proceed as follows: Compute the smallest i ∈ N with S i > n and check whether n is enumerated by g before stage S i .If not, then since by Claim 5 we have g(t) ≥ S i > n for all t ≥ S i , we must have n / ∈ B. ♦ Claim 9a.The sequence (S i ) i is computable in A. Proof.To determine S i for any i ∈ N it suffices to use oracle A to compute the smallest t such that A ∩ {0, . . ., i − 1} ⊆ {f (0), . . ., f (t − 1)} and to output Proof.We claim that there is a recursive algorithm using oracle B that computes (S i ) i .By construction, we have S 0 = 0.So suppose that for i ∈ N the numbers S 0 , . . ., S i are known.We claim that (i) if S i ∈ B and if the uniquely determined (cf.Claim 7) number t with g(t) = S i satisfies f (t) = i and t ≥ S i , then S i+1 = s i+1 [t + 1]; and that (ii) S i+1 = s i+1 [S i ] holds otherwise.Assuming this claim, oracle B clearly computes S i+1 .To see (i), assume that S i ∈ B and that the uniquely determined t with g(t) = S i satisfies f (t) = i and t ≥ S i .By Claim 4 and because f is injective, for all t ′ ≥ t + 1 we obtain f (t ′ ) ≥ i + 1.This implies then there must exist a number t ≥ S i with s i+1 [t + 1] = s i+1 [t] and hence with i + 1 > f (t).Then, by Claim 4, f (t) = i, and by Claim 3, This concludes the proof that for every c.e. set A ⊆ N there exists a regainingly approximable set B ⊆ N with A ≡ T B.
Corollary 19.There is a high regainingly approximable set.

Complexity of Regainingly Approximable Numbers
Let Σ := {0, 1}, let Σ n denote the set of finite binary strings of length n, and write Σ * for n Σ n .For a binary string v, let |v| denote its length.By 0 n we denote the string of length n that consists of n zeros, and for any infinite binary sequence A and any number n ∈ N, by A ↾ n we denote the string A(0) . . .A(n − 1) of length n.For a function for every w ∈ Σ * .If dom(f ) has the property that no two different elements in it can be prefixes of each other then we say that f has prefix-free domain; and in this case it is customary to write K f for C f .Let C denote C U 1 for some optimally universal Turing machine U 1 and let K denote K U 2 for some optimally universal prefix-free Turing machine U 2 (see, for instance, Downey and Hirschfeldt [4, Sections 3.1 and 3.5] for a discussion of such Turing machines).In the remainder of this section we will write U for U 2 .The functions C and K are called the plain and the prefix-free Kolmogorov complexity, respectively.An infinite binary sequence A is called i.o.K-trivial if there exists a constant c ∈ N such that, for infinitely many n, Recalling the conventions laid out in Section 2, we can make the following observation.
Proof.Assume w.l.o.g. that α is not computable and α ∈ (0, 1).Let (a n ) n be an increasing, computable sequence of w.l.o.g.strictly positive rational numbers converging to α such that, for infinitely many n, α − a n < 2 −n .For every n ∈ N, let u n be the binary string of length n with 0.u n ≤ a n < 0.u n + 2 −n .
If u n = 1 n let v n ∈ Σ n be such that 0.v n = 0.u n + 2 −n ; otherwise let v n := u n .Clearly, (u n ) n and (v n ) n are computable.Define a computable function f : dom(f ) → Σ * with prefix-free domain dom(f ) ⊆ Σ * as follows: If w ∈ Σ * satisfies U(w) = 0 n , for some n ∈ N, If at a stage t it looks as if that has happened, we change the weight to a new, smaller value that is at most 2 −t .As all new terms that are added to the series after t will now be scaled by this new weight, their total sum cannot exceed 2 −t .Doing this for infinitely many t then ensures regaining approximability of α.One issue with this approach is that we can never be sure about the stages when we change weights.This is because prefix-free Kolmogorov complexity is uncomputable, and we have to work with its approximations.Thus, it may turn out that what looked like an initial segment of high complexity at stage t really has much lower complexity, and as a result we may have dropped the weight too early.However, since α only needs to be left-computable, it is easy to deal with this: we simply retroactively scale up the weights of all terms that had been given too small a weight.We now come to the formal construction, which will proceed in stages to define computable functions ℓ, r : N 2 → N and w : N 2 → N ∪ {∞} as well as a computable sequence (a t ) t of dyadic rational numbers in the interval [0, 1).We shall write ℓ(n)[t] for ℓ(n, t), r(n)[t] for r(n, t), and w(n)[t] for w(n, t).At stage 0 define ℓ(n)[0] := 0 and r(n)[0] := n and w(n)[0] := ∞, for all n ∈ N, as well as a 0 := 0. At a stage t with t > 0 first define ℓ(0)[t] := 0 and, for n > 0, Note that this is well-defined because for a fixed t we have K(v)[t] = ∞ for almost all v ∈ Σ * .Intuitively speaking, ℓ(n)[t] is our guess at stage t about the length of what we hope will be the n-th witness for the existence of infinitely many initial segments of α that have high complexity.If there is no i < t with ℓ(i for every n ∈ N. Otherwise let i t denote the minimal such i and define for every n ∈ N. In either case, using the convention 2 −∞ := 0, define Using the terminology from the informal proof sketch above, r is employed to reduce weights when we believe that we discovered an initial segment of high complexity, with the intent of ensuring regaining approximability of α.This is then used to define the function w determining the scaling factors that will be applied to the elements of the series n 2 −|h(n)| ; note how the appearance of i t in the definition of w enables retroactively increasing these factors later if required.This ends the description of the construction; we proceed with the verification.It is easy to see that ℓ, r, w, and (a t ) t are computable; we need to argue that α := lim t→∞ a t exists with the desired properties.
The following claims are immediate.
♦ Claim 4. For every n ∈ N the function t → w(n)[t] is nonincreasing.♦ Due to Claim 3, using the convention 2 −∞ = 0, we can write for all t ∈ N. By Claim 4, (a t ) t is nondecreasing, and since for all t ∈ N, its limit α exists and is a left-computable number in (0, 1].
(iii) for every t ≥ t 0 and every i < n 0 , ℓ(i for all n < t 0 and all t ≥ t 0 .
Proof.Fix any n < t 0 and t > t 0 ; in order to show w(n we may inductively assume that the statement has already been proven for t − 1, that is, that w This is clear by construction if there is no i < t with ℓ(i)[t] = ℓ(i)[t − 1]; thus let us assume that such an i does exist.In this case, by construction, w(n Similarly, (ii) and (iii) imply i t 0 = n 0 .Then using Claims 1 and 2 as well as the induction hypothesis we obtain which concludes the proof.♦ Claim 6.For every n ∈ N, the function t → ℓ(n)[t] is eventually constant.Proof.For the sake of a contradiction, assume that there is a smallest n 0 ∈ N such that t → ℓ(n 0 )[t] is not eventually constant.By the definitions above we must have n 0 > 0. Define and, for j > 0, The sequence (t j ) j is trivially increasing and w(n)[t j ] = w(n)[t 0 ] for all n < t 0 and all j ∈ N, by Claim 5. We show by induction that w(n)[t] ≥ r(n 0 )[t 0 ] for all n ≥ t 0 and t ≥ t 0 : • For t = t 0 this is clear by Claim 3.
• For t > t 0 and if there is no i < t with ℓ(i)[t] = ℓ(i)[t − 1] then the assertion is clear inductively by definition of w. • If t > t 0 and there is an i < t with ℓ(i can be assumed to hold inductively and, on the other hand, by Claims 1 and 2 together with the fact that i t ≥ n 0 by choice of t 0 .By choice of t 0 we have for every j ∈ N that i t j = n 0 and thus Combining this with the previous results and using the definition of w, for j > 0 we see that for all t > S and i < t we have ℓ(i for all n ∈ N and all t > S. We obtain a t = a t−1 for all t > S, hence, α = a S .Thus α would be a rational number, which contradicts ( * ). ♦ Consider an m ∈ N with s m > 0. We claim that α − a sm < 2 −sm .
By Claim 4, for every n ∈ N, the function t → w(n)[t] is eventually constant.Thus we can define W n := lim t→∞ w(n)[t] for every n ∈ N and write Proof.By definition of s m , we have ℓ(i for all t ≥ s m and all i < i sm .Thus, the claim follows from Claim 5 with s m in place of t 0 and i sm in place of n 0 .♦ Claim 9.For n ≥ s m we have W (n) ≥ s m .
Proof.It suffices to prove that w(n)[t] ≥ s m for all t ≥ s m and n ≥ s m .We proceed by induction over t.By Claim 3 the assertion is clear for t = s m .Thus consider an arbitrary t > s m .If there is no i < t with , then the assertion is clear inductively by definition of w.So let us assume that there exists such an i.In case that n ≥ t we obtain w(n)[t] = ∞ > s m , and we are done.Otherwise, if n < t, then by definition w(n ).Assuming inductively that the assertion is already proven for t − 1, it is thus enough to show r(i t )[t] ≥ s m .By the definition of s m we have i t > i sm , thus by Claim 2, completing the proof of Claim 9. ♦ Using Claim 8 it follows that and thus, using Claim 9, we can conclude that

Complexity of Regainingly Approximable Sets
As every c.e. set is i.o.K-trivial by a result of Barmpalias and Vlek [2, Proposition 2.2], this holds in particular for every regainingly approximable set.Thus, in analogy to the results in the last section, it is natural to wonder if a regainingly approximable set may at the same time have infinitely many initial segments of high Kolmogorov complexity.
Of course, in the setting of c.e. sets, there is no hope of achieving complexities as high as in the previous section; this is because for any such set A we trivially have C(A ↾ n) ≤ + 2 log n for all n.Kummer [7, Theorem 3.1] showed that there exist c.e. sets that achieve this trivial upper bound infinitely often; to be precise, that there exists a c.e. set A and a constant d ∈ N such that, for infinitely many n, Such sets have been named Kummer complex (see, for instance, Downey and Hirschfeldt [4,Section 16.1]).We will show that there are Kummer complex regainingly approximable sets.For prefix-free complexity, analogous observations can be made: In this setting it is easy to see that for every computably enumerable set A we have K(A ↾ n) ≤ + 2 log(n) + 2 log log(n) for all n ∈ N. Barmpalias and Downey [1, Theorem 1.10] constructed a c.e. set A and a number d ∈ N with K(A ↾ n) ≥ 2 log(n) + log log(n) − d for infinitely many n ∈ N; for the purposes of this article we will call such sets Kummer K-complex.Again, we will show that there are Kummer K-complex regainingly approximable sets.Thus, in this sense, regainingly approximable sets can achieve both maximal plain and maximal prefix-free complexity infinitely often.
Theorem 22.There exists a regainingly approximable set that is Kummer complex.
Corollary 24.For any c.e. sets A, B ⊆ N there exists some d Proof of Theorem 22.Let Z ⊆ N be a c.e. Kummer complex set.By Theorem 15 there exist regainingly approximable sets X, Y ⊆ Z with X ∪ Y = Z.Fix a constant d ∈ N that is large enough to witness the Kummer complexity ( †) of Z and such that ( ‡) is true for all n.If we let c := 2d, then we have, for the infinitely many n satisfying ( †), that , and at least one of X and Y has to be Kummer complex.
By applying the same proof to a set Z that is Kummer K-complex instead of Kummer complex and by replacing the use of Lemma 23 by that of Corollary 24, we can also obtain the following result.
Theorem 25.There exists a regainingly approximable set that is Kummer K-complex.

Arithmetical Properties
The class of regainingly approximable sets is not closed under union, according to Theorems 15 and 17.The following limited closure properties do hold, however, and will be useful in the proof of the next theorem.
(1) The union of a regainingly approximable set and a decidable set is regainingly approximable.(2) If A is a regainingly approximable set and f : N → N is a computable, nondecreasing function, then the set is regainingly approximable.
Proof.Let A ⊆ N be a regainingly approximable set.By Theorem 13 there exists a computable id N -good enumeration g : N → N of A. For the first assertion, let B ⊆ N be a decidable set.Then the function h : N → N defined by h(2n) := g(n) and h(2n + 1) := n + 1 if n ∈ B, h(2n + 1) := 0 if n ∈ B, is a computable and 2n-good enumeration of A ∪ B.
is not regainingly approximable.To see this, let g : N → N be defined by g(n) = ⌊n/2⌋ for all n ∈ N. We observe C = g(A ∩ B).Thus, if A ∩ B were a regainingly approximable set, then so would be C according to Lemma 26 (2), a contradiction.
We return to the study of regainingly approximable numbers.

Corollary 28.
(1) There exists a strongly left-computable number that is not regainingly approximable.
(2) There exists a strongly left-computable number that is regainingly approximable but not computable.
Proof.The first assertion follows from Theorem 17 and Theorem 13.
The second assertion follows from Corollary 16 and Theorem 13 and from the well-known fact that, for any A ⊆ N, the number 2 −A is computable if and only if A is decidable.
From now on, let ≤ S denote Solovay [16] reducibility (see, for instance, Downey and Hirschfeldt [4, Section 9.1]) between left-computable numbers.The following result states that the regainingly approximable numbers are closed downwards with respect to this reducibility.
Proposition 29.Let β be a regainingly approximable number, and let α be a left-computable number with α ≤ S β.Then α is regainingly approximable as well.
Proof.Let f : {q ∈ Q : q < β} → Q be a computable function and c ∈ N be a number such that, for all q ∈ {q ∈ Q : q < β}, we have f (q) < α and α − f (q) < 2 c • (β − q).By Proposition 4 there exists a computable and increasing sequence (b n ) n of rational numbers converging to β such that β − b n < 2 −n−c for infinitely many n ∈ N. The sequence (a n ) n defined by is a nondecreasing, computable sequence of rational numbers converging to α.For the infinitely many n with β − b n < 2 −n−c we obtain Corollary 30.
(1) If the sum of two left-computable numbers is regainingly approximable, then both of them are regainingly approximable.(2) The sum of a regainingly approximable number and a computable number is again a regainingly approximable number.
Proof.The first assertion follows from Proposition 29 and from the fact that for any two left-computable numbers α, β one has α ≤ S α + β and β ≤ S α + β.Since adding a computable number to a left-computable number does not change its Solovay degree, the second assertion follows from Proposition 29 as well.
Corollary 31.Every strongly left-computable number can be written as the sum of two strongly left-computable numbers that are regainingly approximable.
Proof.This follows from Theorem 15 together with Theorem 13.
Corollary 32.There exist two strongly left-computable and regainingly approximable numbers whose sum is not regainingly approximable.
Proof.According to Corollary 28 (1), there exists a strongly left-computable number γ that is not regainingly approximable.Then, according to Corollary 31, there exist two strongly left-computable and regainingly approximable numbers α, β with α + β = γ that witness the truth of the assertion.
Corollary 31 raises the question whether every left-computable number can be written as the sum of two regainingly approximable numbers; the answer is no.This follows from Proposition 20, from the fact that there exist Martin-Löf random left-computable numbers, and from the result of Downey, Hirschfeldt, Nies [5,Corollary 3.6] that the sum of two left-computable numbers that are not Martin-Löf random is again not Martin-Löf random.

Splitting Regular Reals
In Theorem 15 we have seen that for any c.e. set C ⊆ N there exist two disjoint, regainingly approximable sets A, B ⊆ N with C = A ∪ B.
That result left open whether this splitting can be obtained effectively.We now give a positive answer by presenting an according algorithm in the following lemma.We will formulate this splitting algorithm in a more general form so that we may apply it not only to the class of c.e. sets, but in fact to a larger class of objects, namely to the regular reals as defined by Wu [18].For a function f : N → N and a set S ⊆ N from now on we will write Lemma 33.There is an algorithm which, given a function f : N → N with the property that f −1 {n + 1} is finite for every n, computes two functions g, h : N → N such that for every n and such that there exists an increasing sequence (S i ) i with g −1 {1, . . ., S i } ⊆ {0, . . ., S i − 1} for all even i and h −1 {1, . . ., S i } ⊆ {0, . . ., S i − 1} for all odd i.
Proof.Let a function f as in the statement be given.We will describe an algorithm that works in stages to define the desired functions g and h, as well as a function s : N × N → N. We write s i [t] for s(i, t) to express the intuition that s i [t] is our current guess for S i at stage t.
We begin with a high level overview of the proof strategy: Whenever f enumerates a number n at stage t, we need to let either g or h enumerate it.To decide between the two, we follow a greedy strategy in the following sense: for every t the sequence (s i [t]) i will be increasing; if at stage t the largest number i with s i [t] ≤ n is even then we let g enumerate n, otherwise h.As we will show this is sufficient to prove the statement of the lemma; in particular, for every i, (s i [t]) t will turn out to be nondecreasing and eventually constant, allowing us to define With this we are ready to give the formal proof.At stage 0 we define s i [0] := i for all i ∈ N. At a stage t + 1 with t ∈ N we proceed as follows: • If f (t) = 0 then we set g(t) := 0 and h(t) := 0. Intuitively speaking this means that f does not enumerate any new element at stage t, and that we therefore do not let g and h enumerate any new elements either.We also set s i [t + 1] := s i [t] for all i ∈ N. • If f (t) > 0 then this means n := f (t) − 1 is enumerated by f at stage t.We want to let n be enumerated by either g or by h, as follows: If is even then we set g(t) := 0 and h(t) := n + 1 (which means that n is enumerated by h); otherwise, if k t is odd, then we set g(t) := n + 1 and h(t) := 0 (which means that n is enumerated by g).Furthermore, we define s i [t + 1] for all i ∈ N by This ends the description of stage t and of the algorithm; we proceed with the verification.Claim 1.For every t ∈ N, the sequence (s i [t]) i is increasing.
Proof.We prove the claim by induction over t; it clearly holds for t = 0. Fix any t ∈ N and assume that (s i [t]) i is increasing.If f (t) = 0 then (s i [t + 1]) i is identical to (s i [t]) i and hence it is increasing as well.Thus assume that f (t) > 0; then k t is defined by the induction hypothesis and we can observe that (s i [t]) i is increasing: • For any i < j ≤ k t we have • For any i ≤ k t < j we have • For any k t < i < j we have This proves the claim.♦ By Claim 1, for every t ∈ N with f (t) > 0, the number k t is well defined; thus it is clear that g and h as defined by the algorithm satisfy ( * ) for every n ∈ N. It remains to prove the second condition in the statement of the lemma.Claim 2. For every i, the sequence (s i [t]) t is nondecreasing and eventually constant.
Proof: That (s i [t]) t is nondecreasing for every i is clear by definition.We show by induction over i that it is eventually constant as well.It is easy to see that s 0 [t] = 0 for all t.Thus consider an arbitrary number i > 0. By the induction hypothesis there exists a number t 1 such that, for all t ≥ t 1 , s i−1 [t] = s i−1 [t 1 ].Let t 2 be large enough so that t 2 > t 1 and f −1 {1, . . ., s i−1 [t 1 ]} ⊆ {0, . . ., t 2 − 1} (meaning that numbers smaller than s i−1 [t 1 ] are enumerated by f only in stages before t 2 ).Then, for every t ≥ t 2 with f (t) > 0 and for every j < i we have Proof.If this were not true then there would be some t ≥ S i with s i [t + 1] = s i [t], hence, with S i ≥ s i [t + 1] = s i [t] + t + 1 ≥ t + 1 > S i , a contradiction.♦ Claim 4. For every even i, g −1 {1, . . ., S i } ⊆ {0, . . ., S i − 1}.
Proof.Consider an even number i as well as some t ≥ S i with g(t) > 0; we need to show that g(t) > S i .To see this, first observe that the assumption that g(t) > 0 implies that we must have f (t) = g(t) and that k t must be odd.Hence k t = i.If k t were smaller than i then we would obtain s i [t + 1] = s i [t] + t + 1 > s i [t] = S i in contradiction to Claim 3. We conclude i < k t .This implies s i [t] ≤ f (t) − 1 by the definition of k t .As t ≥ S i , using Claim 3 again, we obtain which proves Claim 4. ♦ Claim 5.For every odd i, h −1 {1, . . ., S i } ⊆ {0, . . ., S i − 1}.
Proof.The proof is symmetric to that of Claim 4; it is enough to interchange "even" and "odd" and to replace "g" by "h".♦ As a corollary we immediately obtain the following uniformly effective version of Theorem 15.Note that the algorithm even provides id N -good enumerations of A and B. While we already showed in Theorem 13 that id N -good computable enumerations exist for all regainingly approximable sets, its proof was not fully uniform.Proof.First transform f C into an enumeration without repetitions f C as in Remark 11; then apply the algorithm described in the proof of Lemma 33 to f C .
Lemma 33 can also be applied to regular reals as defined by Wu [18].
(1) For n ∈ N, a real number is called n-strongly computably enumerable (n-strongly c.e.) if it can be written as the sum of n strongly left-computable numbers.(2) If a real number is n-strongly c.e. for some n ∈ N, then it is called regular.
Such numbers can be characterized conveniently by using the following representation δ (in the sense of Weihrauch [17]) of non-negative real numbers: Call f : N → N a δ-name of a non-negative real number α if the series k a k defined by converges to α.Note that, if some f : N → N is a δ-name of a nonnegative real number, then f −1 {i + 1} must be finite for every i ∈ N; if we even have |f −1 {i + 1}| ≤ n for some n ∈ N and all i ∈ N, then we say that f is a (δ, n)-name of α.Then the following lemma is obvious.
Lemma 36.For n ∈ N, a non-negative real number α is n-strongly c.e. if and only if there exists a computable (δ, n)-name f of α.
Applying our previous results to the regular reals, we obtain the following final theorem.
Proof.Applying the algorithm in the proof of Lemma 33 to f γ proves the first assertion.For the second assertion, use the first assertion and the following claim.

( 2 )
(a) ⇒ (b): Suppose that α is computable.Then there exists a computable sequence (b m ) m of rational numbers with |α − b m | < 2 −m for all m ∈ N. Due to (1) it suffices to show that A is computably enumerable.Fix some d ∈ A and define a computable function g : N → N via g( m, n ) := n if b m − a n + 2 −m < 2 −n , d otherwise.Then g computably enumerates A; indeed, if for some m it holds that b m − a n + 2 −m < 2 −n then α − a n < 2 −n ; and for the other direction, if α − a n < 2 −n holds, then for all m satisfying α − a n + 2 • 2 −m < 2 −n we have b m − a n + 2 −m < 2 −n .(b) ⇒ (c): Trivial.(c) ⇒ (a): By assumption A is infinite.Define p A : N → N as the uniquely determined increasing function with p A (N) = A.

Theorem 15 .
For any c.e. set C ⊆ N there exist two disjoint, regainingly approximable sets A, B ⊆ N with C = A ∪ B.Proof.By Sacks' Splitting Theorem[13] (see, for instance, Soare [15, Theorem VII.3.2])there exist two disjoint, c.e., low subsets A, B ⊆ N with C = A ∪ B. Low sets are not high.Hence, by Example 2 and Theorem 13, A and B are regainingly approximable.

Theorem 34 .
Given an enumeration fC : N → N of a set C ⊆ N one can compute enumerations without repetitions f A : N → N of a set A ⊆ N and f B : N → N of a set B ⊆ N such that(1) C is the disjoint union of A and B, and (2) there exist infinitely many t withA ∩ {0, . . ., t − 1} ⊆ Enum(f A )[t]and infinitely many t with B ∩ {0, . . ., t − 1} ⊆ Enum(f B )[t].
If α is rational or if N\A is empty then A and N\A are decidable; thus assume w.l.o.g. that α is irrational and that N \ A is nonempty.Then α − a n ≥ 2 −n if and only if α − a n > 2 −n .Fix any e ∈ N \ A, and define a computable function f and thus s i [t] = s i [t i ] for all t ≥ t i .♦ Define the sequence (S i ) i by S i := lim t→∞ s i [t].By Claim 1, (S i ) i is increasing.Claim 3.For every i ∈ N and every t ≥ S i we have s i [t] = S i .Proof.Assume otherwise and choose t ≥ S i with s hence, k t ≥ i and consequently s i [t + 1] = s i [t].By induction we obtain s i [t] = s i [t 2 ], for all t ≥ t 2 .Thus, (s i [t]) t is eventually constant, and Claim 2 is proven.♦ Let the sequence (S i ) i be defined by S i := lim t→∞ s i [t].Due to Claim 1, (S i ) i is increasing.Claim 3.For every i ∈ N and every t ≥ S i , s i [t] = S i .