MIT OpenCourseWare

Browse materialsMIT 18.100C · Fall 2012

Real Analysis

Subjects: Calculus · Mathematics

Art: Johannes Kepler, Konjukce Saturnu a Jupiteru Originál, 1596 · Public domain

Instructor: Prof. Paul Seidel
Term: Fall 2012
Level: Undergraduate
What's inside: 10 problem sets · 1 exam · 13 solutions · 1 reading · 33 other files · 7 web pages
Materials open: MIT OCW

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Start course: Lecture 1: Sets, ordered sets, countable sets Find a topic

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About this course

Source: MIT OpenCourseWare 18.100C Real Analysis, Fall 2012, Prof. Paul Seidel — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Has MIT solutions 13 files in this course are named by MIT as solutions.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

This course’s cited books, in our reading lists: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics)

How this course was organized

What's here, and what isn't

MIT's own materials here did not support a verified, course-wide order, so they are grouped by section instead.

Not included here

10 items from the source could not be safely included:

MIT's page format for this item wasn't recognized clearly enough to place it safely. (7 items)
Referenced in a table, but without MIT's own verified order, so it isn't used to claim a sequence. (3 items)

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Summaries

25 files

Recitations

Assignments

20 problem sets

Exams

MIT OpenCourseWare, Prof. Paul Seidel, 18.100C Real Analysis, Fall 2012.

License: CC BY-NC-SA 4.0 · Edition: Fall 2012 · Open the original course on OCW ↗

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Real Analysis

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Summaries

Recitations

Assignments

Exams

Lecture 1: Sets, ordered sets, countable sets

Lecture 2: Fields, ordered fields, least upper bounds, the real numbers

Lecture 3: The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz

Lecture 4: Metric spaces, ball neighborhoods, open subsets

Lecture 5: Open subsets, limit points, closed subsets, dense subsets

Lecture 6: Compact subsets of metric spaces

Lecture 7: Limit points and compactness; compactness of closed bounded subsets in Euclidean space

Lecture 8: Convergent sequences in metric spaces; Cauchy sequences, completeness; Cauchy’s theorem

Lecture 9: Subsequential limits, lim sup and lim inf, series

Lecture 10: Absolute convergence, product of series

Lecture 11: Power series, convergence radius; the exponential function, sine and cosine

Lecture 12: Continuous maps between metric spaces; images of compact subsets; continuity of inverse maps

Lecture 13: Continuity of the exponential; the logarithm; Intermediate Value Theorem; uniform continuity

Lecture 14: Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem

Lecture 15: Derivative of inverse functions; higher derivatives, Taylor’s theorem

Lecture 16: Pointwise convergence, uniform convergence; Weierstrass criterion; continuity of uniform limits; application to power series

Lecture 17: Uniform convergence of derivatives

Lecture 18: Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem

Lecture 19: End of Stone-Weierstrass; beginning of the theory of integration (continuous functions as uniform limits of piecewise linear functions)

Lecture 20: Riemann-Stjeltjes integral: definition, basic properties

Lecture 21: Riemann integrability of products; change of variables

Lecture 22: Fundamental theorem of calculus; back to power series: continuity, differentiability

Lecture 23: Review of exponential, log, sine, cosine; e ⁱt = cos(t) + isin(t)

Lecture 24: Review of series, Fourier series