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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.