Problem
ALG-B3-M08-P019 Increasing Cycle of Length Three
#19
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be strictly increasing and suppose \(f^3(x)=x\) for all \(x\). Prove that \(f(x)=x\).
If \(f(a)>a\), the whole chain increases.
If \(f(a)>a\), increasingness gives \(f^2(a)>f(a)>a\), and then \(f^3(a)>f^2(a)>a\), contradicting \(f^3(a)=a\). If \(f(a)
Preparation for the general ordered-cycle fact.