Let \(f_k:[0,1]\to\CC\) be differentiable and suppose \[ |f_k'(x)|\le M \] for every \(k\) and every \(x\in[0,1]\), where \(M\) is independent of \(k\). Show that \(\ts{f_k}\) is equicontinuous.
Hint: MVT.
Solution
For all \(x,y\in[0,1]\) and all \(k\), the mean-value estimate gives \[ |f_k(x)-f_k(y)|\le M|x-y|. \] Hence the sequence is uniformly Lipschitz. Given \(\varepsilon>0\), take \(\delta=\varepsilon/M\) when \(M>0\) (and arbitrary \(\delta>0\) when \(M=0\)). Then \(|x-y|<\delta\) implies \[ |f_k(x)-f_k(y)|<\varepsilon \] for every \(k\), so \(\ts{f_k}\) is equicontinuous.