# Showing a metric space is complete

## Homework Statement

Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric $d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)|$ 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 $d(x_m, x_n)=sup_{0\leq t\leq a}e^{-Lt}|x_m(t)-x_n(t)|$...that's it! I don't know how to find this limit and show that the sequence converges to that.

## Answers and Replies

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.