"(adsbygoogle = window.adsbygoogle || []).push({}); Let f"_{k}be functions defined on R^{n}converging uniformly to a function f. IF each f_{k}is bounded, say by A_{k}, THEN f is bounded.

f_{k}converges to f uniformly =>||f_{k}- f||_{∞}->0 as k->∞

Also, we know|f_{k}(x)|≤ A_{k}for all k, for all x

But why does this imply that f is bounded? I don't see why it is necessarily true.

Any help is appreciated!

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# Uniform Convergence & Boundedness

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