I want to show that $\vert \mathbb{R}^{\mathbb{Z}} \vert = \vert \mathbb{R} \vert $. Do the real numbers and the countable product of real numbers have the same cardinality?
Asked
Active
Viewed 226 times
1
-
Yes. $|\mathbb{R}^\mathbb{Z}| = |\mathbb{R}|^{|\mathbb{Z}|} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0^2} = 2^{\aleph_0} = |\mathbb{R}|$. – user76284 Dec 23 '18 at 06:53