Interactive Real Analysis
- part of Examples 2.1.5:
The set of integers Z and the interval of real numbers between 0 and 2, [0, 2],
are both infinite.
Back
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.
BackDefine 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.