# COMPLETE metric space

For your first question, if n,m>=N, what's the largest value d(n,m) can have? You'd want to take n=N and let m get larger and larger, right? Now just pick that max less than your given epsilon.
If I pick N such that 1/2N-1<epsilon, would this work in the definition of Cauchy?

As for another choice of metric, nothing springs to mind immediately. I do think since the metric space needs to be complete, you need to find a metric that's not totally bounded. Look that up and see if anything comes to mind.
hmm...maybe HallsofIvy can help?

If I define d to be

d(m,n)= 1/2 + ∑1/2k
where the sum is from k=m to k=n-1

will this work or not?

Dick
Homework Helper
Yeah, you've got the right idea on Cauchy. You're new d(m,n) doesn't satisfy d(m,m)=0. Randoming guessing a function isn't going to get you very far. I'll post a note and see if any other homework helpers want to take a crack at this.

In the section of the textbook from which this exercise is taken, the idea of "totally bounded" has not yet been introduced, so I don't think we need this concept. But I do think we need to come up with a metric similar to the discrete metric as suggested by hint 1.

Does anyone have any idea??

Last edited:
Dick