Larry Chan's Blog

Wednesday, September 2, 2026

Math Self-Study Progress Report 2 (DeepSeek Expert vs Gemini Pro)

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.

Monday, August 31, 2026

Real Analysis Metaphor for Dating and Marriage

I was joking around with Gemini while doing real analysis 😂😂😂😂

Click on the image to enlarge it.

Friday, August 7, 2026

Math Self Study Progress Report 1 (Plus Initial Thoughts on AI as Math TA)

For those who are interested, you can read my previous post.

I began self studying real analysis around 2 weeks ago. I've chosen to follow Stephen Abbott's Understanding Analysis based on the comments I read on Amazon.

I have gone through the first chapter, including all the exercises. It covers sets, functions, real numbers, completeness, cardinality and Cantor's Theorem. This is my first time studying real analysis so I have no other books to compare to. But in my opinion, this is an excellent text for self study. I especially like the author's writing and organization. He spends quite some time talking about the motivations behind real analysis and why mathematicians invented it. The proofs were well written, and tend to be on the verbose side of things, but I think it's excellent for first time learners. The exercises reinforce what's being learned and shouldn't be skipped. They force me to think deeper about the material. I also find the epilogue very interesting as it gives readers extra information as well as some historical background regarding the materials covered in the first chapter.

It's only one chapter, but I already feel like it would be a long struggle. Calculus was my strong area when I was in secondary school but this is on another level. Well, technically real analysis NOT calculus! Nothing is intuitive. Definitions and theorems need be read 3-4 times in order to grasp what's being said. All the proofs are like magic. I honestly think that even if I time travelled back to my college days, I would not be spending less effort just because I was younger. So far, I find the Schroder-Bernstein Theorem most impressive and beautiful. I am surprised that its proof is left as an exercise to the reader. It deserves to be fully presented.

I've been using DeepSeek as a studying aid so far. I'm using the free version and I always turn on "expert mode." I don't know the difference between expert mode and think mode and what happens if I turn on both. DeepSeek says they're the same but judging from the output I'm not so sure. Here's how I've been using it.

Verify my answers to the exercises
Ask for hints and directions when I'm stuck in the exercises
Ask for simpler explanations, analogies, and concrete examples to build an intuitive understanding of theorems or definitions
Ask general questions as if it's a real TA (teaching assistant)

Here are my initial thoughts.

One word: INDISPENSABLE. I don't think I would have survived even the first chapter without AI's help.
In particular, as I mentioned above, Schroder-Bernstein Theorem was left as an exercise to the reader. Had DeepSeek not given me a concrete example, I would have never grasped the underlying idea, let alone proving it.
I was saved a number of times by AI when I was really stuck in the exercises. The hints either confirmed that I was thinking in the right direction and motivated me to continue, or gave me a kick in finishing a key step of a proof.
This is specific to DeepSeek. I cannot just take a photo/screenshot of an exercise and ask it to read the question for me. I had to type it in. It's tedious for things like subscripts and set operators. Especially with the latter, I had to describe them in words like "A is a subset of B intersect C."
On the other hand, I was quite surprised that DeepSeek was able to understand what I wrote ALL THE TIME, like a(n) was immediately recognized as "a sub n"; R, N, Q were immediately understood as set of real numbers, natural numbers and rational numbers respectively. At one point, I needed to explain my solution by describing a binary tree with only words. Nevertheless, it was able to grasp my idea correctly and draw that tree out for me. Sometimes I was afraid it might not understand what I said, like "infinite union of A(n)", but so far it never failed to understand me.
It had small misses but they're relatively easy to spot.
There were one or two big misses though, like when it thought a certain mapping is onto but actually not. In both cases, I just asked "Are you sure?" and it was able to recover from its mistakes.
It felt good when it praised the answers I gave or even when I asked questions 😆 For example, "This is a fantastic observation", "Your solution is more elegant than the standard approach." etc.
This is specific to DeepSeek: It couldn't graph functions in-place. It just gave me python code to run. On the other hand, both Gemini and ChatGPT can do that.
This is also DeepSeek specific. I don't understand the difference between Expert mode vs toggling the "Think" button. I asked DeepSeek to explain itself and it said they're the same. But by toggling on the "Think" button, I can see how DeepSeek is thinking. Expert mode does not show how it thinks. So, I'm not sure if they're really the same.

I am pleasantly surprised that AI can now do college level math. In my opinion, it's still more like a TA than a professor. But it's already quite impressive. It's good news for people like me who want to self study. I'm sure it's going to bring huge changes to college education in the near future.

What I'm planning to do is to use Gemini and DeepSeek alternatively for each chapter going forward. This way, I would be able to see how a frontier US model compares to a Chinese frontier model. I thought ChatGPT may be more relevant but I didn't have good experience with it. If I ask too many questions ChatGPT will throttle me. Gemini doesn't seem to have this problem and so I chose it instead.

My Math Self-Study Journey

I have always been interested in math since secondary school. In college, I studied electrical engineering but thought about switching to computer science because that would allow me to double major in math. I eventually abandoned that idea and completed my engineering degree. However, I always regret about my decision since I ended up working as a software engineer anyway.

I retired 5 years ago and the thought of self studying math came to my mind. Based on my own understanding, the bar of getting a math degree in an American college is actually not that high, compared to, say, my homeland Hong Kong. First of all, math majors in Hong Kong would have completed the equivalent of Calculus II in the US when they graduated from secondary school. Therefore, first year math majors in Hong Kong would start with real analysis right away. Contrast with the US, a majority of math majors here would spend the first 3-4 semesters studying calculus and linear algebra, and only began real analysis when they're junior, 2 years later than math majors in Hong Kong. To obtain a math degree at my college, University of Wisconsin-Madison, only 2 upper level math courses, Advanced Calculus I (real analysis) and Advanced Algebra I (abstract algebra), were hard requirements and the rest fell into the electives category. So, a math graduate in US may know nothing about complex analysis or topology, which would never happen in Hong Kong, at least not in my time.

But I digress.

Given my age and my lonnnnnnnnnnnnng hiatus from math, I wanted to set a relatively small achievable goal for myself. Therefore, I used the two hard requirements for math majors at my alma mater as a reference, and set out to self study real analysis as well as abstract algebra. Three years ago, I went through the abstract algebra YouTube series by Michael Penn. To be honest, I already forgot most of it. But it doesn't matter. It's for my own interest. I'm perfectly satisfied with the fact that I once studied it and understood it (isn't this true for most of what we studied in school anyway?). I also enjoyed the experience, and even the frequent struggles, when going through the materials because I find them beautiful.

Recently, inspired by Field Medalists Wang Hong and Deng Yu, I picked up where I left off 2 years ago, and began self studying real analysis. The difference this time is that AI has become much more powerful and I want to use it as a studying aid. In other words, AI would be my math TA (teaching assistant). I'll be reporting my progress, albeit somewhat irregularly. In particular, I would be sharing my experience with using AI as a tool for self studying college level math.

Stay tuned.

August 2026

P.S. I'm not trying to bash or look down on US math education. In my opinion, its math research is world class. I'm just trying to point out the difference between US and Hong Kong regarding math at the undergrad level. I genuinely believe that the hurdle is higher in Hong Kong than US in terms of graduate requirements for math majors. I'm not comparing the quality of the people or the universities.