Why does an odd number plus one, not necessarily entail it being even? Why does an odd number plus one, not necessarily entail it being even?
For example, $\sqrt{5} + 1$ is not even.

An odd number is one of the form $2n+1$ where $n$ is an integer. If $m$ is odd, then $m +1 = 2n+1 +1 = 2(n+1)$ for some integer $n$, and so $m+1$ is even.
As indicated in the comment, $\sqrt{5}$ is not an integer, so the designation of odd/even does not apply.
