Sequence that is Cauchy, bounded, in a complete metric space but does not converge.

Question

I have been stuck on this one for a long time. I need to find a sequence in a complete metric space E such that it is bounded and Cauchy and not convergent. This is kind of tricky. Since E is complete, you just can't have your sequence converge to something outside of E. Since the sequence must be bounded, you can't have a Cauchy sequence that just grows infinitely. I think the sequence must be alternating but I cannot find or come up with any examples of a sequence that is alternating and Cauchy at the same time. Any ideas?

Answer 1

By definition, every Cauchy sequence in a complete metric space converges. The sequence you are looking for doesn't exist.

Answer 2

Firstly, a Cauchy sequence is automatically bounded.

Secondly, a complete metric space is one where, by definitions, every Cauchy sequence converges, so a sequence you search for does not exist.