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Math Help - Uniform Continuous and Uniform Convergence

  1. #1
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    Uniform Continuous and Uniform Convergence

    24.13 Prove that if (fn) is a sequence of uniformly continuous functions on an interval (a,b), and if fn -> f uniformly on (a,b), the f is also uniformly continuous on (a,b). Hint use eps/3 argument.

    I don't see what I need to change from the "limit of continuous functions is continuous" theorem. At the end of that proof it concludes f is continous at x0. I guess I'm not sure how to expand that to uniform continuity.

    Thanks
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  2. #2
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    If \varepsilon  > 0 then \left( {\exists N} \right)\left( {\forall x \in (a,b)} \right)\left[ {n \ge N \Rightarrow \quad \left| {f_N (x) - f(x)} \right| < \frac{\varepsilon }{3}} \right] from uniform convergence.

    Now from uniform continuity \left( {\exists \delta  > 0} \right)\left[ {\left| {x - y} \right| < \delta  \Rightarrow \quad \left| {f_N (x) - f_N (y)} \right| < \frac{\varepsilon }{3}} \right].

    \begin{array}{l}<br />
  \\ <br />
 \left| {f(x) - f(y)} \right| \le \left| {f(x) - f_N (x)} \right| + \left| {f_N (x) - f_N (y)} \right| + \left| {f_N (y) - f(y)} \right| \\ <br />
 \end{array}
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