Sequences; Limits, Limit Supremums; Uniform (/Absolute/Lipschitz) Continuity; Compact and Connected Sets, Metric Spaces
This course will provide you with the rigorous mathematics that underlies calculus. You will see proofs of results you have seen in undergraduate calculus and be introduced to much deeper notions, such as compact sets, uniform continuity, Lipschitz continuity, limit supremum, etc., all in the generality of metric spaces as well.
What you’ll learn
- the rigorous mathematics behind undergraduate calculus and learn new deeper concepts.
- to prepare for Qualifying (a.k.a. Prelim) Exams in graduate programs, e.g. in USA institution.
- to write rigorous and accurate mathematics proofs using proper style and conventions.
- to combine different ideas to produce solutions by doing actual exam questions.
Course Content
- The Real Numbers –> 5 lectures • 50min.
- Sequences (also in Metric Spaces) –> 7 lectures • 55min.
- Series, of Real or Complex Numbers –> 8 lectures • 37min.
- Limit and Continuity at a Point –> 4 lectures • 42min.
- Continuous Functions and Compact Sets –> 9 lectures • 1hr 1min.
- Continuous Functions and Connected Sets –> 6 lectures • 57min.
- Uniform Continuity, Absolute Continuity, Lipschitz and Hölder Continuity –> 6 lectures • 40min.
Requirements
This course will provide you with the rigorous mathematics that underlies calculus. You will see proofs of results you have seen in undergraduate calculus and be introduced to much deeper notions, such as compact sets, uniform continuity, Lipschitz continuity, limit supremum, etc., all in the generality of metric spaces as well.
By working on difficult homework questions, chosen from actual exams, you will be ready to take Qualifying, a.k.a. Prelim, exams in graduate schools. This course can also help with GRE in math subject.
You will learn how to write accurate and rigorous proofs. I will show you how to approach a problem and how to bring together scattered observation to formulate a proof. Then, via examples that I do myself, I show you how to present your solutions in a coherent and rigorous way that will meet the standard expected of graduate students (in qualifying exams).
In this part 1, we cover
- sequences, and their limits (in metric spaces)
- limit supremum and limit infimum
- continuity and semi-continuity
- series, convergence tests
- topological definition of continuity
- continuous functions on compact sets
- continuous functions on connected sets
- local properties
- modulus of continuity
- uniformly continuous functions, Lipschitz and Hölder maps, absolutely continuous functions
A possible future course will cover single-variable differential and integral calculus: uniform convergence of sequences and series of functions, equi-continuity, power series, analytic functions and Taylor series. A whole separate course (or even two) is needed to cover multivariable calculus — but that is far into future.
Here is how material is organized:
- Video Lectures. Each section (7 total) begins with lectures that cover definitions, provide key examples and counter-examples, and present the most important theorems, accompanied with proofs if the proofs are instructive.
- Homework sets with prelim questions. The first lecture of each section contains a downloadable PDF containing question from past prelim exam (or at a comparable difficulty level to them). This is your homework! You are encouraged to spend as much time as you can to try to find solutions on your own.
- Solutions to select exercises from homework sets. In the last video lectures of each section, I show solutions to the *-ed exercises from the homework. I do not just give you a final clean solution. Instead, I walk you through the (initially messy) process that leads one to discover a solution. I teach you how to make small observations and then bring them together to form a solution. Finally, I show you how to write math in a rigorous way that will meet standards of exams and of mathematical conventions. This latter skill cannot be overlooked.
I am so excited to have this course out and cannot wait for your feedback.