Let $m$ and $n$ be positive integers. Prove that if $m \mid n$ where $m < n$ ($m$ is a proper divisor) then $m \leq \frac{n}{2}$.
We're given $m \mid n$ so, for some integer $k$
$$ n = km $$
We're also given $m < n$, so $k>1$. Since $k$ is an integer, this also means $k \ge 2$.
$$ 2 \leq k $$
We can multiply both sides by $m$ since it is given as a positive integer,
$$ 2m \leq km = n $$
Dividing both sides by 2 gives us the desired result,
$$ m \leq \frac{n}{2} $$