Another fun meeting. Matt walked us through his proof in Lean of Exercise 1. He was getting stuck at some point near the end of the proof and we were able to find a slight error in an earlier part of the proof that got him unstuck.
Ed showed us his colorful, visual proof of Exercise 4, which he mailed out above.
Then I walked us through exercises 5, 8, and 6a.
We were really scratching our heads over 6b. That’s where you have to prove that the product of two Cauchy sequences is a Cauchy sequence. I think I just came up with the insight to solve the problem.
In 6a, we had to prove that the sum of two Cauchy sequences is Cauchy. When we were given an epsilon, we used the trick of dividing it in half. That is, show that sequence p converges to within a distance of epsilon/2, and sequence q converges to a within distance of epsilon/2. Therefore the sum converges to a distance within epsilon.
We were stumped on 6b because we can’t just take the square root of epsilon. But there’s no need. All you have to do is take the fact that p converges to within a distance of 1, and q converges to within a distance of epsilon. Multiply them, and it converges to within a distance of epsilon.
Maybe. I have to try to work out the proof.
- Lyle