Examples 2.1.5:

The set of integers Z and the interval of real numbers between 0 and 2, [0, 2], are both infinite.
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According to Dedekind's theorem, we need to find a proper subset of each set that has the same cardinality as the original set. In other words, we need to find a bijection from the original set into a proper subset of itself.

Define the function f(n) = 2n. Then this function is a bijection between Z and the even integers. Hence, Z has the same cardinality as a proper subset of itself, and therefore Z is infinite.

Define the function f(x) = x / 2. Then the function is a bijection between the interval [0, 2] and the interval [0, 1]. Hence, the interval [0, 2] is infinite.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007