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Math 575: Principles of Analysis
The syllabus for the course is available here.
Problem Set 1. Due September 1.
Problem Set 2. Due September 8.
Problem Set 3. Due September 15.
Problem Set 4. Due September 25.
Problem Set 5. Due September 29.
The first midterm and solutions.
Problem Set 6. Due October 20.
Problem Set 7. Due October 27. In problem 4 you should switch the words "maximum" and "minimum"; you may also assume that f'' is continuous at x. Problem 5 should read "...f(x) greater than or equal to 1".
Problem Set 8. Due November 6.
The second midterm and solutions.
Problem Set 9. Due November 20.
Problem Set 10. Due December 1.
A sample exam.
Some notes on the natural numbers.
Some notes on the "continuum".
Some notes on calculus.
Some notes on sequences and series.
Worksheets for Aug. 30, Sept. 8, (with an answer), Sept. 20,Oct. 2, Oct. 20 (with a proof of L'Hopital's rule), and Oct. 30.
As new/aspiring graduate students, you should try to attend seminars. There is some thoughtful advice on attending seminars from Ravi Vakil available here, (skip to the section on general advice), including the three things exercise. Another unrelated but potentially useful read for a beginning mathematician from Math with Bad Drawings, although this is mostly relevant later in your career.