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18.1002

Real Analysis

3-0-9 units · Graduate

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A , for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Students in Course 18 must register for undergraduate version 18.100B .

Spring 2025

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.

Readings

  • Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner. 2008. Elementary Real Analysis . 2nd ed. ISBN: 9781434843678
    Listed reading
  • A screen-optimized PDF version of the textbook is available. ↗
    Listed reading
  • A print-optimized PDF version of the textbook is available. ↗
    Listed reading
  • Rudin, Walter. 1976. Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) . 3rd ed. McGraw-Hill. ISBN: 9780070542358
    Listed reading
  • The latest edition is the 1976 third edition. It’s not available to purchase as an ebook, but it has been digitized by the Internet Archive . You can access the book by creating a free personal Internet Archive account. You can then log in and borrow the book (for one hour at a time, looks like ). ↗
    Listed reading
  • Readings are assigned in Thomson, Brian S., Judith B. Bruckner, and Andrew M. Bruckner. 2008. Elementary Real Analysis . 2nd ed. ISBN: 9781434843678
    Listed reading
  • Lec #: 1 · Lecture Title: Introduction to Real Numbers · Readings: Sec. 1.1–1.4
    Reading assignment
  • Lec #: 2 · Lecture Title: Introduction to Real Numbers (cont.) · Readings: Sec. 1.5–1.6
    Reading assignment
  • Lec #: 3 · Lecture Title: How to Write a Proof; Archimedean Property · Readings: Sec. 1.7
    Reading assignment
  • Lec #: 4 · Lecture Title: Sequences; Convergence · Readings: Sec. 2.1–2.2, 2.4–2.7
    Reading assignment
  • Lec #: 5 · Lecture Title: Monotone Convergence Theorem · Readings: Sec. 2.8, 2.9, 2.11
    Reading assignment
  • Lec #: 6 · Lecture Title: Cauchy Convergence Theorem · Readings: Sec. 2.12
    Reading assignment
  • Lec #: 7 · Lecture Title: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series · Readings: Sec. 2.11–2.12, 3.4
    Reading assignment
  • Lec #: 8 · Lecture Title: Convergence Tests for Series; Power Series · Readings: Sec. 3.5–3.6
    Reading assignment
  • Lec #: 9 · Lecture Title: Limsup and Liminf; Power Series; Continuous Functions; Exponential Function · Readings: Sec. 2.13, 10.2, 5.4, 5.5
    Reading assignment
  • Lec #: 10 · Lecture Title: Continuous Functions; Exponential Function (cont.) · Readings: Sec. 5.5, 5.9
    Reading assignment
  • Lec #: 11 · Lecture Title: Extreme and Intermediate Value Theorem; Metric Spaces · Readings: Sec. 5.4, 13.1–13.2
    Reading assignment
  • Lec #: 12 · Lecture Title: Convergence in Metric Spaces; Operations on Sets · Readings: Sec. 13.4–13.5
    Reading assignment
  • Lec #: 13 · Lecture Title: Open and Closed Sets; Coverings; Compactness · Readings: Sec. 13.5
    Reading assignment
  • Lec #: 14 · Lecture Title: Sequential Compactness; Bolzano–Weierstrass Theorem in a Metric Space · Readings: <none>
    Reading assignment
  • Lec #: 15 · Lecture Title: Derivatives; Laws for Differentiation · Readings: Sec. 7.2, 7.3
    Reading assignment
  • Lec #: 16 · Lecture Title: Rolle’s Theorem; Mean Theorem; L’Hôpital’s Rule; Taylor Expansion · Readings: Sec. 7.6, 7.11–7.12
    Reading assignment
  • Lec #: 17 · Lecture Title: Taylor Polynomials; Remainder Term; Riemann Integrals · Readings: Sec. 8.6
    Reading assignment
  • Lec #: 18 · Lecture Title: Integrable Functions · Readings: Sec. 8.3
    Reading assignment
  • Lec #: 19 · Lecture Title: Fundamental Theorem of Calculus · Readings: Sec. 8.3
    Reading assignment
  • Lec #: 20 · Lecture Title: Pointwise Convergence; Uniform Convergence · Readings: Sec. 9.2–9.4
    Reading assignment
  • Lec #: 21 · Lecture Title: Integrals and Derivatives under Uniform Convergence · Readings: Sec. 9.5–9.6
    Reading assignment
  • Lec #: 22 · Lecture Title: Differentiating and Integrating Power Series; Ordinary Differential Equations (ODEs) · Readings: Sec. 9.5–9.6
    Reading assignment
  • Lec #: 23 · Lecture Title: Existence and Uniqueness for ODEs: Picard–Lindelöf Theorem · Readings: Sec. 13.11.4
    Reading assignment

Prerequisites & course information

Subject meets with 18.100B Prereq: Calculus II (GIR) G (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1001 , 18.100A , 18.100B , 18.100P , 18.100Q

Sources