Question 134
Let $f: \mathbb{Z} \to \mathbb{Z}$ be an injective map such that $|f(m) - f(n)| \le 2015$ holds for arbitrary integers $m$ and $n$ which satisfy $|m - n| \le 2015$. Prove that
$$
|f(m) - f(n)| = |m - n|
$$
holds for all $m, n \in \mathbb{Z}$.
$$
|f(m) - f(n)| = |m - n|
$$
holds for all $m, n \in \mathbb{Z}$.