The Borel-Cantelli Lemmas, and Their Relationship to Limit Superior and Limit Inferior of Sets (or, Can a Monkey Really Type Hamlet?)

The purpose of this chapter is to show that if a monkey types infinitely, Shakespeare’s Hamlet and any other works one may wish to add to the list will each be typed, not once, not twice, but infinitely often with a probability of 1. This dramatic fact is a simple consequence of the Borel-Cantelli lemma and will come as no surprise to anyone who has taken a graduate-level course in Probability. The proof of this result, however, is quite accessible to anyone who has but a rudimentary understanding of the concept of independence, together with the notion of limit superior and limit inferior of a sequence of sets.