Lecture 1 (Sep 7)
We discussed some of the goals of the course, particularly the idea of building up real numbers using only the notion of sets. We reviewed various set operations and tried to imagine what it might mean to define functions purely in terms of sets.
Click here for lecture notesLecture 2 (Sep 11)
We reviewed ways to generate new sets out of given one(s). We discussed an open conjecture about sets, and some recent progress. We revised our definition of ordered pair. We came up with a formal definition of function, and defined a bunch of related concepts.
Click here for lecture notesLecture 3 (Sep 14)
We started our project of defining real numbers via properties. We came up with 11 properties, which, while still not uniquely defining the reals, did eliminate a lot of possibilities. Next time we'll introduce just two more axioms that will uniquely identify the reals!
Click here for lecture notesLecture 4 (Sep 18)
Today we discussed the distributive property (A11), the order property (A12), and their consequences. We proved that 0 multiplied by anything must yield 0 (and we also showed that distributive is necessary to prove this), and also the fundamental result that 1 > 0. Next time we will introduce the final property of ℝ.
Click here for lecture notesLecture 5 (Sep 21)
We discussed property (A12) a bit more, in particular proving some basic properties of inequalities using it. We also proved that the integers (mod 2) don't satisfy (A12). Finally, we thought about what distinguishes ℝ from ℚ; this led us to come with a (final!) axiom, (A13).
Click here for lecture notesLecture 6 (Sep 25)
We continued our discussion of (A13), the completeness axiom. In particular we saw why it's useful to distinguish ℝ from ℚ and how it implies the existence of the infimum of a set. We then switched gears to rigorously constructing some famous subsets of ℝ, like the integers ℤ, the positive integers ℤpos, and the rationals ℚ; This involved coming up with the notion of a successor set. We then proved that induction is a valid proof approach, and practiced using induction by proving two fundamental properties of the integers: that 1 is the smallest positive integer, and that ℤpos is closed under addition.
Click here for lecture notesLecture 7 (Sep 28)
We discussed Strong Induction and used it to prove that the positive integers are well-ordered. Next we tackled an important result: we can approximate any real number pretty well by integers. We proved most of this, and will finish next class.
Click here for lecture notesLecture 8 (Oct 2)
We completed our proof from last time on approximating an arbitrary real number by an integer. This led us to prove the Archimedean Property: that there exist arbitrarily large integers (and therefore, arbitrarily small reciprocals of integers). Finally, we proved half of our first real analysis theorem: that the square-root of 2 is a real number.
Click here for lecture notesLecture 9 (Oct 5)
We completed the proof that the squareroot of 2 is a real number. Then we proved that between any two reals there's always a rational number.
Click here for lecture notesLecture 10 (Oct 12)
We launched a discussion of how to compare infinite sets; are some infinities larger than others? In particular, we discussed Cantor's innovative approach to answering such a question.
Click here for lecture notesLecture 11 (Oct 16)
We continued our discussion of Cantor's fascinating world of sets. We came up with formal and rigorous notions of finite and infinite set, indicated why the set of all rationals has the same size as the set of all positive integers, and began discussing the idea that some infinite sets are strictly larger than others (for example, the unit interval is strictly larger than the set of all positive integers).
Click here for lecture notesLecture 12 (Oct 19)
We proved that the unit interval is uncountable by the diagonal argument, discussed methods for proving bijections between sets (including both visual approaches and the Cantor-Schroeder-Bernstein theorem), and sketched an argument that the length of ℚ is 0 while the length of [0,1] is 1. We concluded with Cantor's proof that any set is strictly smaller than its power set.
Click here for lecture notesLecture 13 (Oct 23)
We finished out our discussion of Cantor's theory of infinite sets with a sketch of the proof that the powerset of the positive integers has the same size as the unit interval, and then discussing the Continuum Hypothesis and Generalized Continuum Hypothesis. Next, we played a couple rounds of a game called the Game of 15; this led us to the notion of isomorphism, which lies at the heart of problem 7.9 (on this week's problem set). Finally, we turned to our next major topic in the course: sequences and limits. We write down a specific sequence, and easily guessed its limit. However, despite many inventive attempts, we were unable to come up with a suitable definition of what we actually mean that this sequence tended to this limit; all our proposals implied other statements we didn't think were true. Next class, we'll start by giving a proper definition.
Click here for lecture notesLecture 14 (Oct 26)
We formally defined the notion of limit (for a sequence), and explored some examples and non-examples. In particular, we introduced a simple but fundamental tool: the triangle inequality.
Click here for lecture notesLecture 15 (Oct 30)
We proved that limits (if they exist) are unique, and discussed the algebra of limits. As an example case, we proved that the limit of a product is the product of the limits.
Click here for lecture notesLecture 16 (Nov 2)
We explored how limits interact with other properties of ℝ, namely order and completeness. In particular, we proved the squeeze theorem and the monotone convergence theorem.
Click here for lecture notesLecture 17 (Nov 6)
We introduced the notion of Cauchy sequence, stated the Cauchy criterion, and demonstrated its utility by considering several examples.
Click here for lecture notesLecture 18 (Nov 13)
We proved the Cauchy criterion, and discussed how it can be used to construct ℝ out of ℚ.
Click here for lecture notesLecture 19 (Nov 16)
We invented the notion of a metric, and explored a bunch of examples of metric spaces.
Click here for lecture notesLecture 20 (Nov 20)
We continued our discussion of metric spaces, in particular discovering and formulating various topological notions (open sets, closed sets, boundary, etc).
Click here for lecture notes Click here a video recording of classLecture 21 (Nov ??)
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Click here for lecture notes*Lecture 24 (Nov ??)
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