A set is countably infinite, if it is equipolent (has the same cardinality), as the set of natural numbers. Countable infinity is the smallest infinity, meaning that every infinite set has a countably infinite subset. The set of real numbers is not countably infinite, which is usually proved by contradiction, if we assume, that there exist a surjection from the set of natural numbers to the set of real numbers. Beside the classic proof, there exist an alternative proof with the help of an infinite real number game. Two players choose some subset S of interval [0,1], and then they alternate choosing real numbers. The first player chooses any real number a_1 between 0 and 1. The second player then chooses any real number b_1 between a_1 and 1. In round n the first player chooses any real number a_n, which satisfies the condition a_(n-1)≤a_n≤b_(n-1), and then the second player chooses number b_n, so that a_n≤b_n≤b_(n-1). The first player has a winning strategy, if he can, without considering the other player strategy, choose the numbers so that α=lim┬(n→∞)〖a_n 〗 is in the set S (because every ascending sequence of real numbers, which is limited above, has a limit). If set S is equivalent to the interval [0,1], the first player has winning strategy, but it is harder to see, that the first player doesn't have a winning strategy, if the set S is countable.
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