Chapter 2 Report
I've just finished studying Chapter 2 of Stephen Abbott's Understanding Analysis. It covers sequences and series, convergence, limits, Bolzano-Weierstrass Theorem, Cauchy Criterion and some more. For me, the most satisfying result is how absolute convergence guarantees that you'll arrive at the same limit no matter how you perform your summation. I don't remember any of my secondary school teacher ever mentioning that order of summation sometimes matters. And I only knew about this for the first time when I was watching a math related YouTube video, may be 10-20 years ago. The Bolzano-Weierstrass Theorem was quite shocking. I guess subsequence was never in the A-Level syllabus so I never gave any thoughts about it.
What's most interesting in this chapter, in my opinion, is when Abbott talks about the relationship between Axiom of Completeness (AoC), Archimedean Property (AP), Nested Interval Property (NIP), Monotone Convergence Theorem (MCT), Bolzano-Weierstrass (BW), and Cauchy Criterion (CC). If you assume AP, all the others are equivalent and you can assume one and derive all others. I have to admit that I didn't give much thoughts when I first read it. But as I read it again and again, its deepness started to sink in. My math teacher in Form 6-7 once told me that being a math major in college, one of the hardest things is to be able to immediately appreciate the theorems proved in class. This is because you'll get indigestion easily when you go to multiple math classes during the day. I think I now understand what he was saying.
There's an unexpected bonus while I was studying Chapter 2 - I picked up LaTeX. I have updated the theme of this blog so that I can embed LaTeX in my blog posts. 😁
DeepSeek vs Gemini
For the first few sections (2.1-2.5) in Chapter 2, I used both DeepSeek Expert and Gemini Pro. I was curious how each responds to the same question. I've reached the conclusion that I'll be using Gemini Pro from now on. In fact, I have been using Gemini exclusively starting from Section 2.6. I didn't pay for Gemini Pro but I got higher usage limits because I pay for Google Drive. But for $20 a month, I would be willing to pay if I were to use it for my self study. Yes, it's that good.
First of all, DeepSeek gives answers quicker than Gemini. But it also made more mistakes. Gemini Pro is very deliberate. It would think for 10-15 seconds before answering but it's rarely wrong. I also like Gemini Pro's writing better. It feels more like human writing, whereas DeepSeek's writing feels a little bit "stiff" and a bit "cold." The only problem I have with Gemini is that it's praising me too much whenever I have the correct answer. It's okay to do it occasionally but there's no need to do it every time.
I have yet to catch a mistake from Gemini. One time, I was asking whether my algorithm works and it said no. But when I pointed out that it misunderstood what I was saying, it was able to recover. I believe there was another time when I thought a bound can be relaxed (e.g. from < to <=) and Gemini only agreed when I pointed it out explicitly. On the other hand, DeepSeek feels a little bit rough on the edges.
To illustrate the difference between DeepSeek and Gemini, here's an exercise I was doing. Click on the images to enlarge since they're hard to see within the web page. Basically, it's about proving Bolzano-Weierstrass (BW) directly by defining a set based on the bounded sequence $(a_{n})$. Solving this amounts to coming up with an algorithm that picks out a subsequence that converges to the least upper bound of this newly defined set, thus proving BW.
I was naively thinking that if $(a_{n})$ is bounded, i.e. $|a_{n}| < M$ for all $n$, then there's always infinitely many terms to choose from in the interval $[-M, s)$, where $s$ is the least upper bound of the set we define. And I thought (again naively) I can then repeatedly halve the interval and still end up with infinitely many terms each time. Thus, at each step, I can pick out a term within that interval to form the desired subsequence.
Here's the answer I gave to Gemini.
As you can see, it immediately pointed out my error, and was able to give me a counterexample right away.
Here's the same answer given to DeepSeek.
DeepSeek got it wrong. It made the same mistake I did, that the sequence is always approaching $s$ from below, and therefore concluded that the interval $[-M, s)$ contains infinitely many terms!
Afterwards, it recovered after I challenged him. It actually gave the same counterexample as Gemini did.
Conclusion
I did the whole Chapter 1 with DeepSeek and it was good. As I said before, I wouldn't have survived this self-study without it. But after going through Chapter 2 with both DeepSeek and Gemini, I prefer the latter. Gemini Pro is simply better.
Next is Chapter 3, which talks about compact sets.
