Proof by contrapositive: if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd
Proof by contrapositive
Contrapositive form:
if the sum of $a$ and $b$ is not odd then $a$ and $b$ are not consecutive integers
I am struck here, so if $a + b$ is not odd means $a + b$ are even
$a + b = 2p$, where $p\in\mathbb Z$.
What are the next steps to show $a$ and $b$ are not consecutive?

WLOG $a=k$ and $b=k+1$ for some natural $k$.
Hence the sum 
$$a+b=2k+1$$
Since $2k$ is always even $a+b$ is always odd
