Math 8365 Spring 2025

Contents (tentative)

1. The Set (N) of Natural Numbers (Sec. 1)
2. The Set (Q) of Rational Numbers (Sec. 2)

1. The Set (R) of Real Numbers (Sec. 3)
2. The Completeness Axiom (Sec. 4)

1. The Symbols (+\infty) and (−\infty) (Sec. 5)
2. Limits of Sequences (Sec. 7)

1. A Discussion about Proofs (Sec. 8)
2. Limit Theorems for Sequences (Sec. 9)

1. Monotone and Cauchy Sequences (Sec. 10)
2. Midterm 1

1. Subsequences (Sec. 11)
2. Series (Sec. 14)

1. Alternating Series and Integral Tests (Sec. 15)
2. Continuous Functions (Sec. 17)

1. Properties of Continuous Functions (Sec. 18)
2. Uniform Continuity (Sec. 19)

1. Limits of Functions (Sec. 20)
2. Power Series (Sec. 23)

1. Uniform Convergence (Sec. 24)
2. Midterm 2

1. More on Uniform Convergence (Sec. 25)
2. Differentiation and Integration of Power Series (Sec. 26)

1. Weierstrass’s Approximation Theorem (Sec. 27)
2. Basic Properties of the Derivative (Sec. 28)

1. The Mean Value Theorem (Sec. 29)
2. Taylor’s Theorem (Sec. 31)

1. The Riemann Integral (Sec. 32)
2. Properties of the Riemann Integral (Sec. 33)

1. Fundamental Theorem of Calculus (Sec. 34)