Last Thursday we ended class with beginning to prove Theorem 2.6.4 which states that a sequence converges is Cauchy. We picked up on Tuesday with a bit of review of this proof: We are assuming is Cauchy and wanting to prove that converges. Using Lemma 2.6.3 and Bolzano – Weierstress we concluded that there exists a convergent sub-sequence such that . From here Dr.LeCrone brought us through his thought process and ultimately showed the class how to piece together the remainder of the proof. He did so in the following steps:

1. What is our claim?

2. What are the knowns? is Cauchy,

3. What do we WTS? such that

4. What will the outline of the proof look like?

– Let …

– Choose N as already given by Cauchy condition

– Let , thus we have $|a_n – a| < \epsilon$

5. Now we need to do some of our side work and fill in the blanks between this proof. I encourage you to go over the class notes to see exactly how Dr.LeCrone did this. We basically used the triangle inequality to expand what we knew and from there fixed a where we got these values from our expansion using the inequality.

Overall, he walked us through building this important proof and showed us the consequences this proof gave:

1. diverges is not Cauchy

2. The negation of Cauchy is a much nicer way of proving divergence

We are now done with sequences and now onto series!

The remainder of class was spent going over Theroems 2.7.2 and 2.7.3 which again I encourage you to look over.