Thread: Are the set of integers not complete?

Define the metric spaces with , the set of integers with the metric

Find sequence in this metric space such that it is cauchy but do not converge.

I understand I must find a sequence that takes only integer values but converge to non-integer. But how should I get it to be cauchy?

Thank you.

By setting Cauchy constant you easily see how all Cauchy sequences in this space must look like. And to what values they converge.

So suppose that is a cauchy sequence in , then , there exists , we have

Set , we then have

Am I on the right track here?

Originally Posted by tttcomrader

So suppose that is a cauchy sequence in , then , there exists , we have

Set , we then have

Am I on the right track here?

It is not possible for unless because are integers and so their difference is always an integer. Thus, the only way you can have an integer less than 1/2 is when that integer is zero.