I would appreciate it if someone could explain the steps in the reasoning of the following statement. This is not a homework assignment or anything.(adsbygoogle = window.adsbygoogle || []).push({});

Let ##U \subseteq R^{n}## be compact and ##f:U\to R## a continuous function on ##U##. However f is not uniformly continuous.

Then there exists an ##\epsilon>0## and sequences ##a_n## and ##b_n## such that if ## || a_n - b_n || < \frac{1}{n} ## then ##||f(a_n) - f(b_n) \ge \epsilon##.

This statement will lead to proof that continuity on a compact interval means uniform continuity. However this is not proven yet so don't use that fact in your reasoning. Purely based on the definitions of uniform continuity.

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# Uniform continuity statement

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