Set of functions from $\mathbb{N}$ to $\mathbb{R}$

Question

How do I prove that the Cardinality of the set of functions from Natural Numbers to Real Numbers is equal to the power set of Real Numbers?

Answer 1

There are $2^{\aleph_0}$ reals, so there are $\left(2^{\aleph_0}\right)^{\aleph_0}=2^{\aleph_0^2}=2^{\aleph_0}$ functions from $\Bbb N$ to $\Bbb R$.