(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Give an example of a decreasing sequence of closed balls in acompletemetric space with empty intersection.

Hint 1: use a metric on N topologically equivalent to the discrete metric so that {n≥k} are closed balls. I_{n}={n,n+1,n+2,...}.

2. Relevant equations

N/A

3. The attempt at a solution

In the following post:

https://www.physicsforums.com/showthread.php?t=374596

We showed that the metric d(m,n)=∑1/2^{k}where the sum is from k=m to k=n-1, satisfies all the conditions required in the problem, except for completeness.

With that metric, we formed the closed balls by taking {n E N: d(k,n)≤1/2^{k-1}} = {k-1,k,k+1,k+2,...} = I_{k-1}. And I_{1},I_{2},I_{3},... is a decreasing sequence of closed balls with empty intersection.

Now, we have to come up with another metric (possibly a modification of the above) that also satisfiescompleteness(i.e. every Cauchy sequence in N converges (in N)).

Does anyone have any idea?

Any help is greatly appreciated!

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# Homework Help: Decreasing sequence of closed balls in COMPLETE metric space

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