US-TST 2023 Problem 6

Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m, n \in \mathbb{N}$ define $$\Delta(m, n)=\underbrace{f(f(\ldots f}_{f(n) \text { times }}(m) \ldots))-\underbrace{f(f(\ldots f}_{f(m) \text { times }}(n) \ldots)) .$$ Suppose $\Delta(m, n) \neq 0$ for any distinct $m, n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exist $m, n \in \mathbb{N}$ with $|\Delta(m, n)|>C$.