TITLE: Does uniform convergence of a sequence of functions imply uniform continuity of its limit?
QUESTION [3 upvotes]: Assume that $f_n$ converges to $f$ uniformly on $C$ and that each $f_n$ is uniformly continuous on $C$. Prove that $f$ is uniformly continuous on $C$.

REPLY [9 votes]: Hint Use this inequality
$$|f(x)-f(y)|\leq|f(x)-f_n(x)|+|f_n(x)-f_n(y)|+|f_n(y)-f(y)|$$
and with the hypothesis of uniform convergence and uniform continity we can make the three terms on the RHS as small as we want.
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