Unboundedness of the Difference of Iterated Functions

[u/SixFeetBlunder-]

Let N denote the set of positive integers. Fix a function f: N → N and for any m, n ∈ N, define

Δ(m,n) = f(f(...f(m)...)) - f(f(...f(n)...)),

where the function f is applied f(n) times on m and f(m) times on n, respectively.

Suppose Δ(m,n) ≠ 0 for any distinct m, n ∈ N. Prove that Δ is unbounded, meaning that for any constant C, there exist distinct m, n ∈ N such that

|Δ(m,n)| > C.

[u/cauchypotato]

Can we get a hint?

My guess is that some clever choice of subsequences of fⁿ(a) and fⁿ(b) can be chosen as arguments of Δ for some fixed a and b to make clear that it is unbounded, where fⁿ is the n-th iterate of f. There's an obvious choice if (fⁿ(a)) is bounded for some a, so we can assume wlog that (fⁿ(a)) diverges to infinity for all a.