### Table of Contents for:

#### Stephen Abbott, Middlebury College

####
Springer-Verlag Press, New York, 2010

## 1. The Real Numbers

#### 1.2 Some Preliminaries

#### 1.3 The Axiom of Completeness

#### 1.4 Consequences of Completeness

#### 1.5 Cardinality

#### 1.6 Cantor's Theorem

## 2. Sequences and Series

#### 2.2 The Limit of a Sequence

#### 2.3 The Algebraic and Order Limit Theorems

#### 2.4 The Monotone Convergence Theorem and a First Look at Infinite Series

#### 2.5 Subsequences and the Bolzano-Weierstrass Theorem

#### 2.6 The Cauchy Criterion

#### 2.7 Properties of Infinite Series

#### 2.8 Double Summations and Products of Infinite Series

## 3 Basic Topology of R

#### 3.2 Open and Closed Sets

#### 3.3 Compact Sets

#### 3.4 Perfect Sets and Connected Sets

#### 3.5 Baire's Theorem

## 4 Functional Limits and Continuity

#### 4.2 Functional Limits

#### 4.3 Combinations of Continuous Functions

#### 4.4 Continuous Functions on Compact Sets

#### 4.5 The Intermediate Value Theorem

#### 4.6 Sets of Discontinuity

## 5 The Derivative

#### 5.2 Derivatives and the Intermediate Value Property

#### 5.3 The Mean Value Theorem

#### 5.4 A Continuous Nowhere-Differentiable Function

## 6 Sequences and Series of Functions

#### 6.2 Uniform Convergence of Sequences of Functions

#### 6.3 Uniform Convergence and Differentiation

#### 6.4 Series of Functions

#### 6.5 Power Series

#### 6.6 Taylor Series

#### 6.7 The Weierstrass Approximation Theorem

## 7 The Riemann Integral

#### 7.2 The Definition of the Riemann Integral

#### 7.3 Integrating Functions with Discontinuities

#### 7.4 Properties of the Integral

#### 7.5 The Fundamental Theorem of Calculus

#### 7.6 Lebesgue's Criterion for Riemann Integrability

## 8 Additional Topics

#### 8.1 The Generalized Riemann Integral

#### 8.2 Metric Spaces and the Baire Category Theorem

#### 8.3 Euler's Sum

#### 8.4 Inventing the Factorial Function

#### 8.5 Fourier Series

#### 8.6 A Construction of R from Q