Teoremas de convergência de sequências de números reais

Abstract

This work aims to present and discuss the main convergence theorems of sequences of real numbers, constituting an essential study for Real Analysis. The first chapter introduces the fundamental concepts of the set of real numbers as a complete ordered field, with emphasis on absolute value (or modulus), set limitation,lower bound, upper bound, infimum and supremum, and characterization of infimumand supremum via the completeness axiom. These concepts provide the theoretical basis necessary for understanding the behavior of numerical sequences. The second chapter systematically develops the theory of convergence of sequences,starting from the definition of limit to demonstrate three classes of fundamental results:operational theorems (uniqueness of limit, algebra of limits), convergence theorems (Comparison, Monotone

4 Convergence, Bolzano-Weierstrass), and the Cauchy Criterion as a characterization of completeness. Each theorem is accompanied by illustrative examples and counterexamples. Methodologically, we adopt a deductive approach based on [[2], Elon Lages Lima, 2006], integrating formal rigor with selected applications that show the relevance of these theorems in the analysis of sequence behavior.