If I pick N such that 1/2N-1<epsilon, would this work in the definition of Cauchy?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.
hmm...maybe HallsofIvy can help?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.
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?