Problem

Source: Serbia TST 2024, P4

Tags: algebra



Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a bijection and let $k$ be a positive integer such that $|f(x+1)-f(x)| \leq k$ for all positive integers $x$. Show that there exists an integer $d$, such that $f(x)=x+d$ for infinitely many positive integers $x$.