1 Lecture 1: Introduction to Real Numbers 2 Lecture 2: Introduction to Real Numbers (cont.) 3 Lecture 3: How to Write a Proof; Archimedean Property 4 Lecture 4: Sequences; Convergence 5 Lecture 5: Monotone Convergence Theorem 6 Lecture 6: Cauchy Convergence Theorem 7 Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series 8 Lecture 8: Convergence Tests for Series; Power Series 9 Lecture 9: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function 10 Lecture 10: Continuous Functions; Exponential Function (cont.) 11 Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces 12 Review for 18.100B Real Analysis Midterm 13 Lecture 12: Convergence in Metric Spaces; Operations on Sets 14 Lecture 13: Open and Closed Sets; Coverings; Compactness 15 Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space 16 Lecture 15: Derivatives; Laws for Differentiation 17 Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion 18 Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals 19 Lecture 18: Integrable Functions 20 Lecture 19: Fundamental Theorem of Calculus 21 Lecture 20: Pointwise Convergence; Uniform Convergence 22 Lecture 21: Integrals and Derivatives under Uniform Convergence 23 Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs) 24 Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem 25 Review for the 18.100B Real Analysis Final Exam