Real Analysis

This course gives an introduction to analysis, and the goal is twofold: 1. To learn how to prove mathematical theorems in analysis and how to write proofs. 2. To prove theorems in calculus in a rigorous way. The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riemann integrals. After that, we will move on to differential equations.

Syllabus

Review for Midterm
Review for the Final Exam
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
Lecture 5: Monotone Convergence Theorem
Lecture 6: Cauchy Convergence Theorem
Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
Lecture 8: Convergence Tests for Series; Power Series
Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
Lecture 10: Continuous Functions; Exponential Function (cont.)
Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
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
Lecture 15: Derivatives; Laws for Differentiation
Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
Lecture 18: Integrable Functions
Lecture 19: Fundamental Theorem of Calculus
Lecture 20: Pointwise Convergence; Uniform Convergence
Lecture 21: Integrals and Derivatives under Uniform Convergence
Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs)
Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem