Convergence of Numerical Sequences

This lecture delves into the convergence of numerical sequences, starting with the demonstration of the proposition that a monotonic sequence, whether increasing and bounded or decreasing and bounded, is necessarily convergent. The proof involves showing that a bounded monotonic sequence has a supremum that also serves as its limit. The concept of sequences defined by recurrence is then introduced, with a focus on linear recurrence. The instructor demonstrates how to determine the convergence and limit of such sequences, highlighting the importance of analyzing the coefficients involved. The lecture also covers the notion of subsequences and the strategy of dividing intervals to determine convergence. Examples and visual aids are used to illustrate the theoretical concepts discussed.