We started class on Thursday, as we will every Thursday from now on, with a situation for us to brainstorm what we actually know about the situation and what we don’t know, but wish we did. In this situation, we had a sequence whose even entries are bounded by 10. After sharing our ideas, we had to make sure we did actually know what we thought we knew based on our basic assumptions. We left this exercise with the advice to take baby steps when writing proofs to make sure everything follows clearly from our assumptions and previous proofs.

We moved on to subsequences and proved Theorem 2.5.2 which states that a subsequence of a convergent sequence converges to the same value as the original sequence. Then we used this result to develop a divergence criterion: If two subsequences converge to different values, then the original sequence must diverge. This divergence criterion is must simpler than trying to negate our definition of convergence which is filled with quantifiers.

Next we used a “Picture Proof” to prove the Bolzano-Weierstrass Theorem (Theorem 2.5.5) which states that every bounded sequence contains a convergent subsequence. Then we discussed Cauchy sequences and how they differ from convergent sequences. In Cauchy sequences, the entries are getting closer to each other (“bunches up” on itself) while convergent sequences are getting closer to a limit value (“bunches up” around a fixed value). This Cauchy definition allows us to determine if a sequence converges without having to know what the sequence converges to.

We ended class with a basic structure of proving Theorem 2.6.4 which states a series converges if and only if it is Cauchy. The proof for the forward direction (Sufficiency) involves the mathematician’s favorite trick and the triangle inequality. The reverse direction (Necessity) requires showing that a Cauchy sequence must be bounded followed by the application of Bolzano-Weierstrass to produce a limit point. We still must prove that the sequence itself converges since B-W only says that there exists a convergent subsequence. This led to the ending advice to know what each theorem you are using actually says and to make sure you have actually shown what you intended.