So I'm studying analysis on my own which has the disadvantage that there aren't many people capable of checking proofs in my neighborhood. So if you would be so kind, I would truly appreciate it!

Problem:

Suppose that is a Cauchy sequence. Prove that is a Cauchy sequence.

I came up with two proofs.

Proof 1:

Since is a Cauchy sequence, and the real numbers are complete, for some . Since the limit of a product of convergent sequences is the product of the limits of the convergent sequences, we have

.Since converges to a finite limit, it is a Cauchy sequence. QED.

Proof 2:

Since is a Cauchy sequence, it converges to a finite limit. Since it converges to a finite limit, it is bounded. That is, there exists a number so that for all . Using the triangle inequality, we see that

for all and .

Let be given. Since is a Cauchy sequence, there exists an so that and implies . But this means that

.

Therefore, is a Cauchy sequence. QED.