Showing a metric space is complete

  • #1
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Homework Statement


Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric [itex]d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)|[/itex] is a complete metric space.

The Attempt at a Solution


Spent a few hours just thinking about this question, trying to prove it directly from the definition that says a complete metric space is one where every Cauchy sequence in it has a limit in the space.

I started with an arbitrary Cauchy sequence of functions [itex]d(x_m, x_n)=sup_{0\leq t\leq a}e^{-Lt}|x_m(t)-x_n(t)|[/itex]...that's it! I don't know how to find this limit and show that the sequence converges to that.
 

Answers and Replies

  • #2
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Try to show that your sequence converges pointswise. So show that for every ##t##, the sequence ##(x_n(t))_n## converges to some real number which I denote by ##x(t)##. Then show that ##x## defined like this is continuous and that ##x_n\rightarrow x## in your metric.
 

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