Prove that all cauchy sequences are bounded in a metric space.
Suppose that A is a metric space, with d the metric. And suppose that the sequence {Xn} is a cauchy sequence on A.
I need to prove that it is bounded, meaning that I have to an element and K such that
Now, since the sequence is cauchy I can set There exist N such that , we have
Consider the following:
(true by the reserve triangular inequality)
implies that
Now, since
we would have
Now, let , therefore we have
But I know something is wrong with this proof because I'm confusing myself with the n, are there anything I can do to clear that up?
Thank you very much!