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