That part is certainly correct and well done. What are you doing with the converse?

If is a Cauchy sequence, then you need to find a number to which it converges.

Again I don’t know what theorems you have.

Here is one way to proceed.

1. Every sequence contains a monotone subsequence.

2. Every Cauchy sequence is bounded.

3. Every bounded monotone sequence converges.

4. Prove the limit of that subsequence is the limit of the whole sequence.