Problem:

If { } and { } are Cauchy sequences in , prove that { } and { } are also Cauchy sequences.

You cannot use the Theorem 2.6.4 which states, "Every Cauchy sequence of [tex]\mathbb{R}/MATH] converges.

My work:

We can't use Theorem 2.6.4, but Theorem 2.6.2 says, "Every convergent sequence of is a Cauchy sequence." So in order for me to prove { } and { } are also Cauchy sequences, I need to prove that { } and { } converge.

I found this:

But our professor told us to prove this claim directly by using , and think that proof above uses the theorem 2.6.4 which we are not allowed to use.

My work so far:

Assume { } and { } are Cauchy sequences in . Thus > 0, such that < . And also, > 0, such that < .

Let > 0. Since < , also < . Thus we have + < + = .

This is where I got stuck/confused: How to prove { } converges using epsilon because when I multiply , I get lost because I'm seeing too many different subscripts . How do I go about finishing the proof using epsilons to proof convergence so I can say the sequences are Cauchy sequences?

Any help, suggestions, corrections, and tips are greatly appreciated.

Thank you for your time.