MathLabs

Problem 6

Let f:Z>0→Z>0f:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}. Prove that if f(n+1)>f(f(n))f(n+1)>f(f(n)) for every positive integer nn, then f(n)=nf(n)=n for every positive integer nn.
Step 5 of 5: Conclude
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In plain words

Matching upper and lower bounds force equality.

m≤f(m)≤m⟹f(m)=mm\le f(m)\le m\Longrightarrow f(m)=m
Detailed analysis

The lower bound from positivity and strict increase and the upper bound just proved coincide. Therefore f(m)=mf(m)=m for every positive integer mm.