As we have stated, proof is central to mathematics; without proof mathematics loses its power. In this section we show how to approach a proof and break it down to manageable, understandable pieces. Important clues and helpful hints as to why a theorem is true are often removed in the final written version of a proof. The construction lines are erased, and, unfortunately, it is up to the reader to reconstruct them. Think of the proof as being a tight bundle you have to unpack.

You may need some more experience to get the full benefit from the following advice; a lot of it should make sense straight away but rereading at a later date could be useful.

A simple theorem and its proof

The following theorem is easy to understand. It has similarities with Theorem 1 in the previous chapter but is different. The subsequent proof will be the primary example in this chapter.

Theorem 18.1

Let m and n be natural numbers. The product mn is odd if and only if m and n are odd.

Does this theorem seem reasonable? Why is it different to Theorem 1 of the previous chapter? Apply the ideas of that chapter to this new statement.