I am supposed to prove that the set of all bounded real functions on a metric space with metric is complete.
So we have some Cauchy sequence . If we fix we can clearly see that is a Cauchy sequence in and thus there exists some such that . So, define by . It is an elementary concept from analysis that a Cauchy sequence of functions converges uniformly. Thus, we can make by making sufficiently large regardless of what value is. Thus, for every there exists some such that and so . So, it remains to show that is bounded. But, this is trivial, since there exists some such that for every and thus but since is bounded for every we see that there exists some such that and thus is bounded and thus it is in our metric space.
So, we may conclude that this space is complete.
How's that look?