# 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 $k$ is an odd positive integer, then $k^{2}$ is also odd.

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 form^{1}^{1}
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

$\displaystyle(2p+1)^{2}$ | $\displaystyle=4p^{2}+4p+1$ | (6.1) | ||

$\displaystyle=2(2p^{2}+2p)+1$ | (6.2) |

As $2(2p^{2}+2p)+1$ is in the form^{2}^{2}
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.

$\Box$