Course

Real analysis

Course under construction · 23 of 32 concepts published; more are added as they are reviewed.

Build calculus rigorously: real numbers, sequences, continuity, differentiation and Riemann integration. It prepares you to read and write proofs and to see why calculus works.

Recommended before this course: Calculus (9 concepts come from that course and are tagged)

32concepts

5topics

410exercises

≈ 18 hof study approx.

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Course path

1
Real numbers 6 concepts
Ordered field · Supremum and infimum · Completeness axiom · Archimedean property · Absolute value · Density of the rationals
2
Sequences 7 concepts
Convergent sequence · Uniqueness of limits · Monotone sequence · Monotone convergence theorem · Subsequence · Bolzano-Weierstrass theorem · Cauchy sequence
3
Limits and continuity 6 concepts
Continuous function from Calculus · Intermediate value theorem from Calculus · Extreme value theorem from Calculus · Limit of a function from Calculus · Sequential continuity · Uniform continuity
4
Differentiation 5 concepts
Derivative from Calculus · Rolle's theorem from Calculus · Mean value theorem from Calculus · L'Hôpital's rule from Calculus · Differentiability and continuity
5
Integration and series 8 concepts
Fundamental theorem of calculus from Calculus · Partition and Darboux sums · Riemann integral · Integrability of continuous functions · Numerical series · Comparison test · Absolute convergence · Uniform convergence

Sources

Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015.
Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011.
James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
Tom M. Apostol, Calculus, Vol. 1, 2.ª edición, John Wiley & Sons, 1967.