Problem 1131 (difficulty: 3/10)

A sequence of functions \(\displaystyle f_1,f_2,\ldots:I\to\R\) is uniformly bounded, if \(\displaystyle \exists K\in\R ~ \forall n\in\N ~ \forall x\in I ~ |f_n(x)|<K\).

Prove that the limit of a uniformly bounded sequence of functions is bounded.

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