I am working on this problem, but I am not sure about one of my arguments. Could someone help me?

Let be a metric space. Define . Prove that is complete iff is complete.

Suppose is complete. Let be a Cauchy sequence in . Let . Then there exists N such that for all then

If , then implies that .

If , then I want to conclude that , but I can't justify this.

So is Cauchy in . Since is complete, converges to some . This implies there exists M such that if , then . Let . Then . So, if then . So, converges to .

I think the converse is analogous, if I can show that any Cauchy sequence in is Cauchy in , then the result would follow.