# Book:Elon Lages Lima/Análise Real 1

## Elon Lages Lima: Análise Real 1

Published $\text {1989}$, Insituto Nacional de Matemática Pura e Aplicada

ISBN 978852440048.

### Contents

Prefácio
1. Finite and Infinite Sets
1. Natural numbers
2. Finite sets
3. Infinite sets
4. Countable sets
5. Exercises
2. Real Numbers
1. $\R$ is a field
2. $\R$ is a totally ordered field
3. $\R$ is a complete totally ordered field
4. Exercises
3. Sequences of Real Numbers
1. Limit of a sequence
2. Limits and inequalities
3. Operations with limits
4. Infinite limits
5. Exercises
4. Series
1. Convergent Series
2. Absolutely convergent series
3. Convergence tests
4. Commutativity
5. Exercises
5. Some Topological Notions
1. Open sets
2. Closed sets
3. Accumulation points
4. Compact sets
5. The Cantor Set
6. Exercises
6. Limits of Functions
1. Definition and first properties
2. Lateral limits
3. Limits at infinity, infinite limits, indetermined expressions
7. Continuous Functions
1. Definition and first properties
2. Continuous functions on an interval
3. Continuous functions on compact sets
4. Uniform continuity
8. Derivatives
1. The notion of derivative
2. Operational rules
3. Derivative and local slope
4. Differentiable functions on an interval
9. Taylor's Formula and Applications of The Derivative
1. Taylor's formula
2. Convex and concave functions
3. Successive approximations and Newton's method
10. The Riemann Integral
1. Revision about sup and inf
2. Riemann integral
3. Properties of the integral
4. Sufficient conditions of integrability
11. Calculus with Integrals
1. The classical theorems of the integral calculus
2. The integral as a limit of Riemann sums
3. Logarithms and exponentials
4. Improper integrals
12. Sequences and Series of Functions
1. Simple convergence and uniform convergence
2. Properties of the uniform convergence
3. Power series
4. Trigonometric functions
5. Taylor series