(5) Bonus question: countability via games. Recall that a set S is countable if there exists a bijection (one-to-one correspondence) f: S N from S to the natural numbers. Equivalently, S is countable if it can be written as S = {$1,S2,...}. Recall also that the interval [0,1] is not countable (Cantor, 1874). We will prove this using a game. This proof is due to Grossman and Turett (1998). Consider the following game. Fix a subset S [0,1], and let ao = 0 and bo = 1. The players Al and Betty take alternating turns, starting with Al. In an-1, but Al's nth turn he has to choose some an which is strictly larger than strictly smaller than bn-1. At Betty's nth turn she has to choose a bn that is strictly smaller than bn-1 but strictly larger than an. Thus the sequence {an} is strictly increasing and the sequence {bn} is strictly decreasing, and furthermore an