The following notes are organized by lecture date. The notes from lecture are handwritten on my iPad and vary in format depending on the content of the lecture. I include questions underneath the relevant lecture notes. Additionally, there is a short summary for each lecture that includes my thoughts on the content and discusses the content in perspective of the entire course.
This lecture is the foundation of the rest of the course. We start from the naturals and build everything in the course off of this first number set. It is quite amazing everything that we can construct from this very first concept, and something that one cannot appreciate until much later in the course. We also introduce induction, which is also a fundamental (and also very cool) concept for constructing proofs, something we do a lot of in this course.
Notes: jan19.pdf
We continue with foundational definitions in this lecture, introducing the idea of min, max, sup, and inf. We also distinguish the reals from the rest of the sets with the Completeness Axiom. These concepts dominate the first half of the course as we begin to analyze sequences and limits.
Notes: jan21.pdf
The limit lecture
In this lecture, we introduce the idea of sequences and limits. The definition of a limit is not something that should be forgotten, since we use this to prove not only all other theorems in this lecture, but it pops up again and again and again through the rest of the course. For me, this was a really interesting lecture as it formalized the idea of a limit, something that has been taught to me again and again using a very hand-wavy explanation. Additionally, this is the first place we show how important epsilon can be when proving ideas, especially when we are talking about anything related to infinity. There's a lot of really interesting things we can prove using epsilon. Using it while proving limits gives us some practice with this idea.
Notes: jan26.pdf
This one takes a bit.
In the first part of this lecture, we go over how to do proofs using the definition of a limit. This takes a little bit to get used to when first learning, but soon becomes fairly straightforward and a lot of the concepts learned here can be applied to questions throughout the course.
In the second part of the lecture, we introduce the idea of monotone sequences, which are very handy as they guarantee convergence and other useful properties. Then, we introduce the idea of limsup and liminf, concepts which seem very foreign at first but give a good introduction to thinking about sequence behavior at infinity. These become very important concepts throughout the course and its important to understand the nuances of these concepts (ex: the difference between limsup and max).
Notes: jan_28.pdf
Cauchy Sequences This lecture we focus on understanding one proof, or rather, 4 proofs as there are three properties which guarantee each other. A cauchy sequence, a convergent sequence, and limsup = liminf. There are quite a few nuances to these proofs, but we do notice some similar themes from the limit proofs in the previous lecture and those from the homework. Rewatching this lecture, these proofs are much easier to understand as I have a much stronger grasp of what these three properties mean.
Notes: feb2.pdf
Subsequences
Just as we can guarantee properties of monotone sequences, once we introduce the idea of a subsequence, we can use these to guarantee new properties.
Notes: feb4.pdf
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Notes: feb16.pdf
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Notes: feb23.pdf
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Notes: mar2.pdf
Notes: april6.pdf