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Is there a relationship between uniform continuity and uniform convergence? For example, suppose $\{f_{n}\}$ is a sequence of functions each of which is uniformly continuous on $[a, b]$. Then does it follow that $f_{n}$ converges to $f$ uniformly on $[a, b]$? (Maybe with some additional conditions?)

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2 Answers 2

No, for example each $f_n$ can be equal to a constant $c_n$, but such that the sequence of real numbers $\{c_n\}$ is not convergent. Even if $\{f_n\}$ converges pointwise, it's not enough (take $f_n(x)=x^n$ on $[0,1]$).

However, it's true that a uniformly limit on $I$ of uniformly continuous functions on $I$ is uniformly continuous on $I$. To see that, use a $3\varepsilon$-argument: take an integer such that the uniform distance between $f$ and $f_n$ is $\leq\varepsilon$, and use uniform continuity of $f_n$ on $I$ to get the result.

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The mere fact that all the functions in a sequence are uniformly continuous can impossibly be enough to show any sort of convergence. Take e.g. $f_n(x)=n$ (constant functions).

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