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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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    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.2016-01-18

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