So I am trying to prove that a sequence is cauchy if and only if it is convergent using only the Bolzano Weierstrass therom for sequences

[Any bounded sequence must have at least one convergent subsequence]

So far, all I have gotten in the forward direction (cauchy => convergent) is that a cauchy sequence is bounded, and hence by the Bolzano Weierstrass theorem, there must be at least ONE convergent subsequence. I have a theorem in the book which states:

[A sequence converges to A iff each of its subsequences converge to A.]

Is this the proper theorem to use in my problem and if so how do I show that ALL of the subsequences converge, or even that more than one subsequence exists at all?

Also in the backwards direction (convergent => cauchy), I am completely at a loss. Any help would be greatly appreciated.

Thanks!