COMPLETE metric space

  • Thread starter kingwinner
  • Start date
  • #26
1,270
0
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?
 
  • #27
Dick
Science Advisor
Homework Helper
26,263
619
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.
 
  • #28
1,270
0
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:
  • #29
Dick
Science Advisor
Homework Helper
26,263
619
You might want to post another thread on this so it will get renewed attention. I'm not seeing it.
 

Related Threads on COMPLETE metric space

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
5
Views
824
  • Last Post
Replies
1
Views
999
  • Last Post
Replies
1
Views
905
Replies
12
Views
2K
  • Last Post
Replies
5
Views
3K
Replies
2
Views
2K
  • Last Post
Replies
3
Views
4K
Replies
5
Views
11K
Top