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