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December 23, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


This course covers the fundamentals of mathematical analysis:

convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in written and oral presentation.

The three options for 18.100:

  • Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible.
  • Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology.
  • Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication. This fulfills the MIT CI requirement.

Course Curriculum

  • Sets, ordered sets, countable sets Unlimited
  • Fields, ordered fields, least upper bounds, the real numbers Unlimited
  • The Archimedean principle; decimal expansion; intersections of closed intervals; complex numbers, Cauchy-Schwarz Unlimited
  • Metric spaces, ball neighborhoods, open subsets Unlimited
  • Open subsets, limit points, closed subsets, dense subsets Unlimited
  • Compact subsets of metric spaces Unlimited
  • Limit points and compactness Unlimited
  • Convergent sequences in metric spaces Unlimited
  • Subsequential limits, lim sup and lim inf, series Unlimited
  • Absolute convergence, product of series Unlimited
  • Power series, convergence radius; the exponential function, sine and cosine Unlimited
  • Continuous maps between metric spaces Unlimited
  • Continuity of the exponential; the logarithm Unlimited
  • Derivatives, the chain rule; Rolle’s theorem, Mean Value Theorem Unlimited
  • Derivative of inverse functions; higher derivatives, Taylor’s theorem Unlimited
  • Pointwise convergence, uniform convergence Unlimited
  • Uniform convergence of derivatives Unlimited
  • Spaces of functions as metric spaces; beginning of the proof of the Stone-Weierstrass Theorem Unlimited
  • End of Stone-Weierstrass Unlimited
  • Riemann-Stjeltjes integral: definition, basic properties Unlimited
  • Riemann integrability of products; change of variables Unlimited
  • Fundamental theorem of calculus; back to power series: continuity, differentiability Unlimited
  • Review of exponential Unlimited
  • Review of series, Fourier series Unlimited
  • Correction Unlimited

About the instructor

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Massachusetts Institute of Technology