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!