Every attempt to give an elementary rigorous course of Analysis is beset by several unsurmountable difficulties. Every teacher of the subject will readily agree that the two main obstacles are (i) Dedekind’s theorem, and (ii) the Heine-Borel theorem. Of these the first may be rendered less apparent, for in an elementary treatment there is no great objection to the postulation of the existence of an upper and a lower bound for every bounded linear set of numbers. The second cannot, however, be avoided, for the important theorems of the differential calculus, such as Rolle’s theorem, the Mean-value theorem, Cauchy’s formula, and Taylor’s theorem can only be rigorously proved by appealing to certain properties of continuous functions, which in their turn depend upon the Heine-Borel theorem, or upon some alternative simpler theorem on intervals such as the one which Carslaw uses to replace it.
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