Check my proof. Please :)

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! (Happy)

** 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

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.

Re: Check my proof. Please :)

Both of your proofs assume that you have a sequence of real numbers, but that isn't specified in the problem.

In proof 2, you only use the completeness of the real numbers to show that the sequence is bounded. I think you can show that without assuming completeness.

- Hollywood

Re: Check my proof. Please :)

Quote:

Both of your proofs assume that you have a sequence of real numbers, but that isn't specified in the problem.

In proof 2, you only use the completeness of the real numbers to show that the sequence is bounded. I think you can show that without assuming completeness.

- Hollywood

Hi Hollywood,

Thank you for taking time to look at my proofs. I really appreciate it.

I guess I should have prefaced the post with the fact that I'm studying **real** analysis, so it is implied that I'm dealing with a sequence of real numbers.

In proof 2, I used completeness to show the sequence converges. Which in turn I used to show the sequence is bounded.