Let \(f_n\in C^1([0,1])\) satisfy \[ f_n(t)\le5, \qquad |f_n'(t)|\le1 \] for all \(n,t\). Define \[ g_n(t)=\max\{f_1(t),\ldots,f_n(t)\}. \] Prove that \((g_n)\) converges uniformly on \([0,1]\).
Solution
Each \(f_n\) is \(1\)-Lipschitz. The maximum of finitely many \(1\)-Lipschitz functions is again \(1\)-Lipschitz, so every \(g_n\) is continuous and \[ |g_n(s)-g_n(t)|\le|s-t|. \] Also \[ g_n(t)\le g_{n+1}(t)\le5 \] for every \(t\). Hence the pointwise limit \[ g(t)=\sup_n g_n(t)=\sup_n f_n(t) \] exists and is finite. Moreover, taking suprema in the Lipschitz estimate gives \[ |g(s)-g(t)|\le|s-t|, \] so \(g\) is continuous.
Thus \((g_n)\) is a monotone increasing sequence of continuous functions on the compact interval \([0,1]\) converging pointwise to the continuous function \(g\). By Dini’s theorem, the convergence is uniform.