Real Number Sequences: Cauchy Sequences

This lecture continues the discussion on real number sequences by introducing the important concept of Cauchy sequences, which allows controlling the existence of a limit without knowing its value. Cauchy sequences help model and understand real numbers as equivalence classes of Cauchy sequences of rational numbers. This modeling is crucial practically, as it explains why real numbers can often be approximated by rationals in scientific calculations. The lecture also covers sub-sequences, the Bolzano-Weierstrass theorem for bounded sequences, and provides an initial overview of numerical series with parameters, which will be further explored in the course as functions defined by power series. Some questions, such as the convergence of certain numerical series, will be addressed later in the course, particularly in relation to integration methods.