converging sequence of continuous functions

Question

Is the following true? 'If a sequence of continuous functions $f_n$ defined in a bounded closed region converges to a continuous function $f$ as n approaches infinity, then $f_n$ uniformly converges to $f$.' If it is, how can I prove it? If it isn't, is there any counterexample?

Answer 1

See Munkres book on Topology. Your statement is true if $f_n$ is additionally a monotone increasing sequence. But in general it doesn't have to be true.

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The following is a counterexample to your statement. Define

$$f_n(x) = \begin{cases} nx, x \in [0,\frac{1}{n}] \\ 1-n(x-\frac{1}{n}), x \in [\frac{1}{n},\frac{2}{n}] \\ 0, [\frac{2}{n},1] \end{cases}$$

$f_n$ is continuous on $[0,1]$ and converges pointwise to the zero-function, but not uniformly.