Because is a Cauchy sequence we have
.
In other words, Cauchy sequences are bounded.
So your sequences are bounded. .
Now,
.
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.