18.100B | Spring 2025 | Undergraduate, Graduate

Real Analysis

Lecture Notes

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18.100B S25 Full Lecture Notes: Real Analysis
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18.100B S25 Lecture 01: Introduction to Real Numbers
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18.100B S25 Lecture 02: Introduction to Real Numbers (cont.)
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18.100B S25 Lecture 03: How to Write a Proof; Archimedean Property
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18.100B S25 Lecture 04: Sequences; Convergence
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18.100B S25 Lecture 05: Monotone Convergence Theorem
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18.100B S25 Lecture 06: Cauchy Convergence Theorem
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18.100B S25 Lecture 07: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series
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18.100B S25 Lecture 08: Convergence Tests for Series; Power Series
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18.100B S25 Lecture 09: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function
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18.100B S25 Lecture 10: Continuous Functions; Exponential Function (cont.)
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18.100B S25 Lecture 11: Extreme and Intermediate Value Theorem; Metric Spaces
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18.100B S25 Lecture 12: Convergence in Metric Spaces; Operations on Sets
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18.100B S25 Lecture 13: Open and Closed Sets; Coverings; Compactness
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18.100B S25 Lecture 14: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space
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18.100B S25 Lecture 15: Derivatives; Laws for Differentiation
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18.100B S25 Lecture 16: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion
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18.100B S25 Lecture 17: Taylor Polynomials; Remainder Term; Riemann Integrals
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18.100B S25 Lecture 18: Integrable Functions
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18.100B S25 Lecture 19: Fundamental Theorem of Calculus
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18.100B S25 Lecture 20: Pointwise Convergence; Uniform Convergence
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18.100B S25 Lecture 21: Integrals and Derivatives under Uniform Convergence
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18.100B S25 Lecture 22: Differentiating and Integrating Power Series; Ordinary Differential Equations
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18.100B S25 Lecture 23: Existence & Uniqueness for ODEs: Picard–Lindelöf Theorem
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308 kB
18.100B S25 Review for Midterm
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18.100B S25 Review for the Final

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