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

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

As $2(2p^{2}+2p)+1$ 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 $2\cdot\text{something}+1$ which is important. $2x+1$, we have shown that this is true.

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