Real Analysis: Lecture Notes

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 cover.pdf cover.pdf Blank Page