Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 22G / b / Solution

Choose $A>0$ and setTakeInductively, if $|f_n(s)|\le A$ on $[0,T_1]$, thenSince $f_0=0$, every iterate is bounded by $A$. Moreover,Thus $(f_n{)}_{n\ge 1}$ is uniformly bounded and uniformly Lipschitz, hence equicontinuous, on $[0,T_1]$.