Book:Elon Lages Lima/Análise Real 1

Subject Matter

 * Real Analysis

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