Suppose this game can continue indefinetely. One of the players has the rationals, the other the irrationals. The one with the rationals chooses a subinterval I1 = [a1 , b1] of [0, 1] (a1 e b1 may be rational or irrational) with length b1 - a1 <= 1 = 1/1. On the next move, his opponent chooses a subinterval I2 = [a2, b2] of I1 with length less or equal to 1/2. On the n-th move, it's chosen a subinterval I_n = [a_n, b_n] of I_(n-1) with length b_n - a_n <= 1/n. So, 2 sequences a_n e b_n are constructed in [0,1], such that a_n < b_n for every n and lim (a_n - b_n) = 0 Since the reals are a complete set, a_n and b_n converge to a same x in [0,1] (ie, x is the only element common to all of the I_n's). If x is rational, the player with the rationals wins, and vice-versa. Show that there exists an strategy that, if followed by the player that has the irrationals, assures his victory no matter the moves of his opponent are.
2007-07-02
11:12:57
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3 answers
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asked by
Steiner
7