Real Analysis I · Continuity
Uniform limits preserve a common Lipschitz bound
Problem
Suppose $f_n:E\to\mathbb R$ are all $L$-Lipschitz for one constant $L\ge0$, and $f_n\to f$ uniformly on $E$. Prove $f$ is $L$-Lipschitz and hence uniformly continuous.
Hint
Fix $x,y\in E$ and compare $f(x)-f(y)$ through one function $f_n$.
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