Corollary. Every infinite set contains a proper subset to which it is similar. Proof. Let A be an infinite set and let S = {a, az, ..} be a count- ably infinite subset of A. The function f: A → A-{a;} defined by (an+1 if x = a, (n = 1, 2, ...), if x € A-S %3D f(x) = (x is a bijection. Thus A ~ A-{a,}. |

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