Lecture 1: Introduction to Real Numbers
Lecture 2: Introduction to Real Numbers (cont.)
Lecture 3: How to Write a Proof; Archimedean Property
Lecture 4: Sequences; Convergence
Problem set 1 due
Lecture 5: Monotone Convergence Theorem
Lecture 6: Cauchy Convergence Theorem
Problem set 2 due
Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
Lecture 8: Convergence Tests for Series; Power Series
Problem set 3 due
Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
Lecture 10: Continuous Functions; Exponential Function (cont.)
Problem set 4 due
Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
Review for Midterm
Problem set 5 due
Midterm exam
Lecture 12: Convergence in Metric Spaces; Operations on Sets
Lecture 13: Open and Closed Sets; Coverings; Compactness
Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space
Problem set 6 due
Lecture 15: Derivatives; Laws for Differentiation
Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
Problem set 7 due
Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
Lecture 18: Integrable Functions
Problem set 8 due
Lecture 19: Fundamental Theorem of Calculus
Lecture 20: Pointwise Convergence; Uniform Convergence
Problem set 9 due
Lecture 21: Integrals and Derivatives under Uniform Convergence
Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)
Problem set 10 due
Lecture 23: Existence and Uniqueness for ODEs: Picard–Lindelöf Theorem
Review for the Final
Final exam
