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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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  • $\begingroup$ As a small addition to the correct answer below: you get uniform convergence to some $f$ if additionally your sequence is bounded, and uniformly equicontinuous (i.e. the delta in the continuity can be chosen the same for all functions of your sequence). This is the Arzela-Ascoli theorem. $\endgroup$ – Bananach Jan 18 '16 at 6:18
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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 uniform 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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