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Theorem. Z the set of all integers is a countably infinite set. Z J f 1 0 n if n is even Proof Define f J Z by f n n 1 f n 2 if n is odd n 1 We now show that f maps J onto Z. Note that 2 w 1 is an odd positive number. So 2 w 1 1 f 2w 1 w. Hence f maps J onto Z. w 0. Let w Z. If w 0 then note that f 1 0. Suppose 2w w. Suppose w 0. Solving w for n we get n 2 w 1.
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