Understand the problem

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

Source of the problem

IMO longlist 1977

Topic
Functional equations
Difficulty Level
Medium
Suggested Book
Functional Equations by B J Venkatachala

Start with hints

Do you really need a hint? Try it first!

Prove (by induction on m) that if n\ge m then f(n)\ge m.
Note that hint 1 implies that f(n+1)>f(n).
Note that hint 2 implies that n+1>f(n), i.e. f(n)\le n.
Hint 3, combined with hint 1 gives f(n)=n for all n\in\mathbb{N}.

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