September 4, 2025 Done with Chapter 7 on series. Finite and infinite series! Wow! A part of an analysis course is now officially covered. Root and ratio test proofs are very dry. I will revisit them again, and perhaps I also need solve some exercise from other books to get some practice finding limits to weird sequences.

Next up: Infinite sets — the last chapter before we embark the journey of behavior/structure on functions. This chapter (8) shall be an important closure to the basics of analysis. We will finally formally deal with infinite sets, countability, union of such sets e.t.c We will also a very fascinating theorem due to Riemann —

An infinite series which is not absolutely convergent can be rearranged to converge to any real number!!!

This result is just crazy.

Also, an important result is that, an infinite series which is absolutely convergent - absolutely converges under any rearrangement! Unlike the finite, case we need something extra for rearranging without doubt!

Infinite series are basically limits of sequences of finites series! Another intuitive result is that

July 26, 2025 Done with Chapter 6! Marks the half way point of this journey! This chapter progress has been slow — mostly because, I started working on Algebra and Topology too, and boy oh boy have I not been using my brain juices on full fledge. This chapter marks a very important stage in my education in real analysis. The defining part of number systems is finally done. From here on, we will probe the reals mostly by studying new objects on them — series, and functions. Each having their own theory.

I am now ready to write the article on these number systems, and tackle the unsettling questions about reals! I think there has been a very good progress in my mathematical maturity too through this chapter. Of course, last chapters were very foundational in this aspect, but this chapter has been an entry to some real (😉) stuff. Of course, so many beautiful things await!

July 21, 2025 Was away from coursework in the weekend. Decided to focus on project like stuff in the weekends. I wrote the $\epsilon$-adherence story clearly — reminisced about the intuition, the proof and the struggle to recall the intuition! But the proof holds everything! That’s the beauty of math. Just pushed it to the GitHub page. Also, updated the Recounting Numbers: A hike to Reals article with some terrain information :). Will continue it in the next weekend. Next up in the Terry Tao’s book is to work on the lim sup, limit point, lim inf exercises. Very nice piece of concepts capturing the upper and lower end behavior of a sequence. Predicted some neat result, and it turned out to be true. The proof is scheduled to be worked out in the coming exercises. Chapter 6 will be done this week, and I will try to complete writing the basic structure and the content of the article. Chapter 6 marks a good closure to Reals (in the sense of defining them and understanding their place — but more generally speaking, we are only beginning to probe more) as indicated before. Last week has been great with the green inking, reading and solving stuff. Have been enjoying my chalk board too :).

July 8, 2025 Chapter on Real Numbers is done! Just like — and // were used to define integers and rationals, we had a placeholder object called $LIM_{n \to \infty}$ : which is associated to (equivalent) Cauchy sequences. Taking examples, we understand what this object would be - 0.9, 0.99, 0.999, 0.9999, ….. The LIM is then just 1, a rational. and 1.4, 1.414, 1.4142,…. LIMITS to 2^0.5. Identifying these LIMITS requires us to define a notation for the reals (just like -n for integers and a/b for rationals). The notation here being the nth roots and decimal system. Of course, these examples are for our guiding intuition, which we use in order to develop the theory which makes sense in its own right. That is, without reference to examples, one can simply work with the abstract object LIM and deduce all the algebra, order relations and exponentiation laws (which directly follows from the rations). Of course one cannot forget the role played by the examples. Examples are our machines to construct abstract theory, the theory then flows far, but when in doubt and need of development it’s again the examples that ground it and take it ahead.

That’s the theme of real numbers (perhaps even the whole of math) in this chapters. Next up, we identify this LIM object as a product of the limit operation $lim$. The abstraction that lingers on when moving from rationals to reals shall fade, but new questions emerge and the story continues - one has to challenge their thinking to facilitate and answer new questions…

A question: In what ways reals are incomplete…. In what sense reals are complete? Of course, I’m tacitly referring to the missing roots of negative reals and completeness of reals.

In this chapter we barely scratched the breadth of real numbers. But yes, the most powerful property reals hold is that of Least Upper Bound property. It single handedly tells us how complete the reals are… I think what I am getting at is: How to classify the LIM objects? nth roots? Decimal system? Are all real but irrational numbers nth roots?