Proving convergent sequences are Cauchy sequences
Solution 1:
Your proof is correct.
Secondly, the property of having every Cauchy sequence converge is very important, and is known as completeness.
As an example, $\mathbb R $ with the usual metric is complete.
Another important area of study is Banach spaces, which roughly are complete metric spaces where the metric comes from a norm.
More generally, there are Hilbert spaces, which are equipped with an inner product.