Mathematical induction. Basic properties of real numbers. Limit of a sequence. Series of real numbers. Absolute and conditional convergence. Main convergence tests for series. Notions of topology on the line. Continuous functions; operations. Intermediate value theorem. Weierstrass theorem on extremes of continuous functions. Uniform continuity. Derivative at a point. Chain rule. Relationship between derivative and growth. Mean value theorem. Convex functions. Integrable functions. Fundamental theorem of calculus. Change of variable. Integration by parts. Average theorem. Taylor’s formula.

References:
LIMA, E. L. – Análise Real, Vol. 1, Rio de Janeiro, IMPA. Coleção Matemática Universitária, 1999.
LIMA, E. L. – Curso de Análise, Vol. 1, Rio de Janeiro, IMPA, Projeto Euclides, 1989.
LANG, S. – Analysis I, Reading, Mass., Addison-Wesley, 1968.
RUDIN, W. – Principles of Mathematical Analysis. 2nd ed., New York, McGraw-Hill, 1964.

* Basic syllabus. The teacher has the autonomy to make any changes.