Mathematical Analysis

Enumerability; concept of infimum and supremum; construction of real numbers. Sequences and numerical series: notion of limit, Monotonous Sequence Theorem, Embedded Intervals Theorem, Cauchy sequence, Bolzano-Weierstrass Theorem, convergence criteria. Line topology: open, closed, convex, connected sets. Real functions. Limit. Continuity: Bolzano's Theorem, Intermediate Values Theorem, Weierstrass Theorem, Uniform continuity. Derivation: monotony, Mean-Value Theorem, L'Hôpital Rules, Taylor, optimality criteria. Riemann integral; Fundamental Theorem of Calculus; Mean-Value Theorem for Integrals. Mention to the integral and Lebesgue measure. Sequences and series of functions: punctual, uniform convergence, limit under the integral sign. 

Basic Information

Workload
45 hours

Mandatory:

  • Lima, E. L. Curso de Análise. Volume 1. Projeto Euclides, IMPA, 2000.
  • Rudin, Walter. Principles of mathematical analysis. Vol. 3. New York: McGraw-hill, 1964.
  • Neri, Cassio e Cabral, Marcos; Curso de Análise Real; IM-UFRJ, 2011.

Complementary: 

  • Thomson, B.S., Bruckner, J.B. and Bruckner, A.M. Elementary real analysis. 2008.
    ClassicalRealAnalysis.com
  • Lages, Elon. Análise real. Volume 1. Coleção Matemática Universitária, IMPA, 1989.
  • Abbott, Stephen. Understanding analysis. New York: Springer, 2001.
  • Apostol, T. M. Mathematical analysis. Addison-Wesley Reading, 1964.
  • Rudin, W., Real and Complex Analysis, McGraw-Hill Book Company Inc., New York-Toronto-London,
    1974.
  • Bartle, R.G., Sherbert, D.R. Introduction to Real Analysis, John Wiley & Sons, Inc., 2011