# Book:Elon Lages Lima/Análise Real 1

## Elon Lages Lima: *Análise Real 1*

Published $1989$, **Insituto Nacional de Matemática Pura e Aplicada**

- ISBN 978852440048.

### Subject Matter

### 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

- 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