Teaching

 
Chapter 1. Finite and infinite sets: natural numbers, finite sets, infinite sets, countable sets.

 Chapter 2. Real Numbers: R is a field, R is an orderly body, R is a complete ordered body.

 Chapter 3. Sequences of real numbers: Limit of a sequence, Limits and inequalities, operations with limits, infinite limits.

 Chapter 4. Numerical Series: convergent series, absolutely convergent series, convergence criteria, Commutativity.

 Chapter 5. Notions of topology: topological spaces, open sets, closed sets, accumulation points, compact sets, Topology on the straight, the Cantor set.

 Chapter 6. Limits of Functions: definition and first properties, lateral limits, limits at infinity, infinite limits, indeterminate expressions.

 Chapter 7. Continuous Functions: Definition and first properties, continuous functions on an interval, continuous functions on compact sets, uniform continuity.

 Chapter 8. Derivatives: The notion of derivative, operational rules (Insist on functions not derivable at a point), derivative and local growth, differentiable functions on an interval.

 Chapter 9. Taylor's formula and applications of the derivative: Taylor's formula, convex and concave functions, successive approximations and Newton's method.

 Chapter 10. The Riemann Integral Review on sup. and inf., Riemann integral, integral properties, sufficient conditions of integrability.

 Chapter 11. Integral Calculus with: the classical theorems of integral calculus, the integral as limit of Riemann sums, logarithms and exponentials, improper integrals.

 Chapter 12. Function sequences and series: convergence simple and uniform convergence, uniform convergence gives properties, power series, trigonometric functions, Taylor series.
 

References

•  Análisis Real, Volumen  1. Elon Lages Lima, Colección Matemática Universitaria. Instituto Nacional de Matemática Pura y Aplicada. Río de Janeiro, Brasil.
• E.L. Lima, Curso de Análisis, Volumen 1. Proyecto Euclides, IMPA, 1994.