Problem
ALG-B3-M08-P020 Any Finite Iteration
#20
★★★★★ Level 5 of 5
Let \(k\ge2\), \(f:\mathbb R\to\mathbb R\) be strictly increasing, and suppose \(f^k(x)=x\) for all \(x\). Prove that \(f(x)=x\) for all \(x\).
Consider three cases: \(f(a)>a\), \(f(a)
Fix \(a\). If \(f(a)>a\), then strict increasingness gives \(f^2(a)>f(a)\), then \(f^3(a)>f^2(a)\), and so on. Thus \(f^k(a)>a\), a contradiction. If \(f(a)
Final problem of the module: general ordered-iteration principle.