Uncountable sets and an infinite real number game

The game. Alice and Bob decide to play the following infinite game on the real number line. A subset S of the unit interval [0, 1] is fixed, and then Alice and Bob alternate playing real numbers. Alice moves first, choosing any real number a1 strictly between 0 and 1. Bob then chooses any real number b1 strictly between a1 and 1. On each subsequent turn, the players must choose a point strictly between the previous two choices. Equivalently, if we let a0 = 0 and b0 = 1, then in round n ≥ 1, Alice chooses a real number an with an−1 < an < bn−1, and then Bob chooses a real number bn with an < bn < bn−1. Since a monotonically increasing sequence of real numbers which is bounded above has a limit (see [8, Theorem 3.14]), α = limn→∞ an is a well-defined real number between 0 and 1. Alice wins the game if α ∈ S, and Bob wins if α 6∈ S.