6.1 Direct proof

A direct proof is probably what you intuitively think of when asked to “prove” something.

Often we try to prove statements that are true for some cases - for example

Example 6.1.1

Show that if kk is an odd positive integer, then k2k^{2} is also odd.

We start by assuming that kk is an odd positive integer. To do this, we want to find an algebraic way to represent kk. As every odd number can be written in the form11 1 This can be proved by induction (see Section 6.3) 2p+12p+1 for suitable pp (e.g. 3=21+13=2\cdot 1+1, 5=22+15=2\cdot 2+1, etc.) we can write kk as 2p+12p+1. From there we apply our assumption to show that the result is true

(2p+1)2\displaystyle(2p+1)^{2} =4p2+4p+1\displaystyle=4p^{2}+4p+1 (6.1)
=2(2p2+2p)+1\displaystyle=2(2p^{2}+2p)+1 (6.2)

As 2(2p2+2p)+12(2p^{2}+2p)+1 is in the form22 2 It’s not the specific expression or variable names which we chose that are important here - it’s the overarching structure of the expression - i.e. that it’s in the form 2something+12\cdot\text{something}+1 which is important. 2x+12x+1, we have shown that this is true.