(Items sorted in reverse chronological order)
2020-11-15 Week 11 (11/16 - 11/20) «Mean Value Theorem and Friends»
This week will focus on the Mean Value Theorem for differentiable functions, and some of its applications.
2020-11-08 Week 10 (11/9 - 11/13) «Differentiability»
This week will focus on differentiability of functions.
2020-11-01 Week 9 (11/2 - 11/6) «Properties of Continuous Functions»
We will talk about first some basic properties of continuous functions, and then move on to the biggies: the Intermediate Value and Extremal Value Theorems and their applications.
2020-10-25 Week 8 (10/26 - 10/30) «Limits of functions; continuity»
We will talk about what it means for functions to have limits, and we will use this definition to define continuity.
2020-10-18 Week 7 (10/19 - 10/23) «More on limsup and liminf; a bit of series.»
The textbook definition of limsup and liminf; arithmetic of limsup and liminf. Series and its convergence. Some basic series convergence tests explained.
2020-10-11 Week 6 (10/12 - 10/16) «Subsequences and Bolzano-Weierstrass Theorem»
In this week, we study the notion of subsequences. A particularly powerful consequence of the completeness of the real number line is the Bolzano-Weierstrass Theorem, which guarantees that any bounded sequence has a subsequential limit.
2020-10-03 Week 5 (10/5 - 10/9) «Cauchy’s Criterion and Monotone Convergence»
The goal of this week are two convergence theorems, first is the Monotone Convergence Theorem, second is the Cauchy Criterion for convergence. In order to make sense of the latter, we will take a quick detour to the concepts of limit superior and limit inferior.
2020-09-27 Week 4 (9/28 - 10/2) «Limits of Sequences»
Definition and properties of limits for sequences.
2020-09-21 Week 3 (9/21 - 9/25) «Supremum and Infimum»
Defining supremum and infimum, how to prove things related to those concepts, consequences of the completeness property of the reals.
2020-09-14 Week 2 (9/14 - 9/18) «Completeness of reals»
We define what it means for the reals to be complete, and contrast it against the incompleteness of the rationals. To do so, we rely on the notion of maximum and minimum elements.
2020-09-06 Week 1 (9/8 - 9/11) «Cardinality, countability, Archimedean property»
As a warm up to analysis, we first discuss the cardinality and countability of subsets of real numbers. Then we move on to basic arithmetic properties of integers, rationals, and reals, and finally get toward the Archimedean property of the rational numbers.
2020-08-31 Week 0 (9/2 - 9/4) «Review and Overview»
Overview of what Analysis I is about, review of concepts that students should already be familiar with.