The direct method of proof is probably the most basic and will be used in the other methods. Despite this there are a couple of pitfalls that are easy to fall into. Two of the most common are assuming what had to be proved and incorrect use of equivalence. We shall investigate these in this chapter. And since it would be nice to gather common mistakes together in one handy chapter rather than having them distributed throughout the book some other mistakes are included. Also brought in is an explanation of why we can't divide by zero – a mistake you probably already know about but may not have been given a reason why. I have seen all these errors made and, like most mathematicians, have made them myself.

Don't assume what had to be proved

Probably the most common mistake in proofs is assuming what had to be proved. Suppose that we had to prove statement P. If we assume it is true, then it is not surprising that we can deduce it is true; P ⇒ P would seem to be very obviously true. Another error in this vein is that P is assumed to be true and this is used to deduce something that is true and so it is concluded that P is true. This is of course an incorrect argument.