Real Analysis

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 through a study of real numbers, and teaches an understanding and construction of proofs.

• Lecture 1: Sets, Set Operations and Mathematical Induction
• Lecture 2: Cantor's Theory of Cardinality (Size)
• Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
• Lecture 4: The Characterization of the Real Numbers
• Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value
• Lecture 6: The Uncountabality of the Real Numbers
• Lecture 7: Convergent Sequences of Real Numbers
• Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences
• Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem
• Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series
• Lecture 11: Absolute Convergence and the Comparison Test for Series
• Lecture 12: The Ratio, Root, and Alternating Series Tests
• Lecture 13: Limits of Functions
• Lecture 14: Limits of Functions in Terms of Sequences and Continuity
• Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function
• Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem
• Lecture 17: Uniform Continuity and the Definition of the Derivative
• Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function
• Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem
• Lecture 20: Taylor's Theorem and the Definition of Riemann Sums
• Lecture 21: The Riemann Integral of a Continuous Function
• Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
• Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions
• Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits
• Lecture 25: Power Series and the Weierstrass Approximation Theorem