Self Study

13 May 2026

Since my last log, I have been continuing with sequences in real analysis. Finally got to the squeeze theorem, it was nice to see an old friend ;(. I really want to get to series, because an integral is a limit of a sum.

6 May 2026

I was doing an exercise on convergence of sequences, and I really liked the wrong answer to this question:

Consider the sequence 0, 1, 0, 0, 1, 0, 0, 0, 1, … with n zeros between the nth 1. Does it converge?

I knew the sequence could not converge because the only numbers in the sequence are 0 and 1, so the limit can only be 0 or 1. But |1 − 0| = 1, so neither can be within every positive ε of the other — a contradiction for any candidate limit.

But then I tried to prove the opposite. If you look carefully, the block [x_n, …, x_n+1] contains 100/n% ones and (1 − 100/n)% zeros. Those percentages themselves converge — the density of 1s tends to 0 — suggesting the sequence converges to 0. This is going in the wrong direction in terms of convergence, but it does prove something cool.

5 May 2026

I have been continuing with real-analysis convergance of sequences. I was so distrubed for a long time about if it is okay to use '>=' in place of > if you are unsure. Yes you can apparently - '>' means '>=', but '>=' does not mean '>'. One is simply a weaker statment.

I forgot how to factorize a cubic polynomial. For like 15 seconds.

4 May 2026

I was focsuing on convergance of sequences in real analysis today. I was suprised to find that I forgot so much about absolute values from first-year. I struggled to get my head around this "algorithm" when it comes to covergance, but I feel I am satified now. A sequence converges when we can find an algorithm for generating an N such that all values bigger than xN are smaller than any epsilon.

3 May 2026

Having finished the very first part of sequences and series of RA, the definitions of infinitely often and eventually have just been presented to me. I really like the formal notation, this is something that was not used in engineering.

So cool to see the axiomatic foundations developed in chapter one being used to show that a sequence converges.

The author briefly remarked a topological view of the definition of convergance, and it is the coolest thing I have seen so far. The contraint of defining distance is abandoned, and all we need is open sets.

2 May 2026

I Finally started with sequences and series in real analysis. The last questions from the first-chapter were quite challenging, and keeping the proofs cogent was hard. I also had to get a feel for when to use which axioms casually without explaining it. I reckoned that I need to focus on being rigorous about the concept being asked about, but should not accidentally derive all of algebra if the question is about completeness. The questions in the beginning of chapter two forced me to remember first-year and high school sequences and series. I was distrubed to find that the rationals are not closed when limits are introduced - i.e. a sequence of rationals can converge to an irrational, that was really cool to see.

24 April 2026

Worked through the completeness axiom and the Archimedean property today. What strikes me is how much weight the field axioms are carrying — I keep finding myself wanting to ask why they are the way they are, rather than simply accepting them as given. This has made me want to look at abstract algebra at some point, just to understand what is truly load-bearing in the structure of ℝ versus what is more contingent.

17 April 2026

Started the Ouwehand notes properly.

10 April 2026

First session. Set up this page. Emailed Professor Taylor, who was kind enough to share the notes used in FTX5058F and confirm that Shreve I and II are the primary references. Beginning with real analysis as a foundation before touching stochastic calculus - the plan is sequences and series first, then continuity, then into Shreve once the ε-δ machinery feels natural.