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18.1001

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. Proofs and definitions are less abstract than in 18.100B . Gives applications where possible. Concerned primarily with the real line. Students in Course 18 must register for undergraduate version 18.100A .

Fall 2020

Topics

  • Sets, Set Operations, and Mathematical Induction
  • Cantor’s Theory of Cardinality (Size)
  • Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property
  • The Characterization of the Real Numbers
  • The Archimedian Property, Density of the Rationals, and Absolute Value
  • The Uncountabality of the Real Numbers
  • Convergent Sequences of Real Numbers
  • The Squeeze Theorem and Operations Involving Convergent Sequences
  • Limsup, Liminf, and the Bolzano-Weierstrass Theorem
  • The Completeness of the Real Numbers and Basic Properties of Infinite Series
  • Absolute Convergence and the Comparison Test for Series
  • The Ratio, Root, and Alternating Series Tests
  • Limits of Functions
  • Limits of Functions in Terms of Sequences and Continuity
  • The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet’s Function
  • The Min/Max Theorem and Bolzano’s Intermediate Value Theorem
  • Uniform Continuity and the Definition of the Derivative
  • Weierstrass’s Example of a Continuous and Nowhere Differentiable Function
  • Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem
  • Taylor’s Theorem and the Definition of Riemann Sums
  • The Riemann Integral of a Continuous Function
  • The Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
  • Pointwise and Uniform Convergence of Sequences of Functions
  • Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits
  • Power Series and the Weierstrass Approximation Theorem
  • Final Assignment

Readings

  • Lebl, Jiří. Basic Analysis I: Introduction to Real Analysis, Volume 1 . CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401.
    Listed reading

Prerequisites & course information

Subject meets with 18.100A Prereq: Calculus II (GIR) G (Fall, Spring) 3-0-9 units Credit cannot also be received for 18.1002 , 18.100A , 18.100B , 18.100P , 18.100Q

Sources