Sequences: Convergence and Divergence

In Section 2.1, we consider (infinite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or negative infinity. We present a number of methods to discuss convergent sequences together with techniques for calculating their limits. Also, we prove the bounded monotone convergence theorem (BMCT), which asserts that every bounded monotone sequence is convergent. In Section 2.2, we define the limit superior and the limit inferior. We continue the discussion with Cauchy sequences and give examples of sequences of rational numbers converging to irrational numbers. As applications, a number of examples and exercises are presented. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.