Analysis Proof Involving Cauchy Sequences

So I teach high school math, but I love to study math in my spare time. I'm charging my way through an analysis book, trying to solve all of the problems. Been looking at this one for a couple of days now and I need to get un-stuck. It's clear to me that and that since , but I'm stuck on proving it. Any help appreciated. Thanks! (Happy)

Here is the problem:

Suppose that and are both Cauchy sequences and that for each n. Prove that

Re: Analysis Proof Involving Cauchy Sequences

Quote:

Originally Posted by

**UNLVRich** Here is the problem:

Suppose that

and

are both Cauchy sequences and that

for each n. Prove that

Because is complete we know that

All you need to show is that .

Have you (can you) shown that if

If you can then let

Re: Analysis Proof Involving Cauchy Sequences

Quote:

Originally Posted by

**Plato** Have you (can you) shown that if

I've boiled it down to being able to show , but there is something that I'm missing. I'm not sure how to show that.

Re: Analysis Proof Involving Cauchy Sequences

is a snake that crawls arbitrarily close to . On the other hand, if , then there is a finite distance between and 0. Eventually the distance between and becomes smaller than this distance. What happens then?

Re: Analysis Proof Involving Cauchy Sequences

Quote:

Originally Posted by

**UNLVRich** I've boiled it down to being able to show

, but there is something that I'm missing. I'm not sure how to show that.

I take you at your word that you are self studying basic analysis. Here is very useful concept.

First, "almost all" means **all but a finite collection**.

So if then almost all of the terms are 'near' .

As was pointed out in reply #4, if that would impossible if

Re: Analysis Proof Involving Cauchy Sequences

Hi,

Maybe this will help. Try proof by contradiction. Namely, assume c < 0. Then in the definition of limit take . Then there is N such that . But this says , a contradiction.

Re: Analysis Proof Involving Cauchy Sequences

Thanks to johng, Plato and emakarov. I will meditate further on this. :)