Problem 1138 (difficulty: 5/10)

A sequence of functions \(\displaystyle f_1,f_2,\ldots:I\to\R\) is uniformly Lipschitz, if \(\displaystyle \exists K\in\R ~ \forall n\in\N ~ \forall x,y\in I ~ |f_n(x)-f_n(y)|\le K|x-y|\). Prove that a pointwise limit of a sequence of uniformly Lipschitz functions is Lipschitz.

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