Question 176
Fix a function $f: \mathbb{N} \to \mathbb{N}$ and for any $m, n \in \mathbb{N}$ define
$$
\Delta(m, n) = \underbrace{f(f(\dots f(m)\dots))}_{f(n) \text{ times}} - \underbrace{f(f(\dots f(n)\dots))}_{f(m) \text{ times}}.
$$
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 $|\Delta(m, n)| \gt C$.
$$
\Delta(m, n) = \underbrace{f(f(\dots f(m)\dots))}_{f(n) \text{ times}} - \underbrace{f(f(\dots f(n)\dots))}_{f(m) \text{ times}}.
$$
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 $|\Delta(m, n)| \gt C$.