I need to prove that these two sets have the same cardinality: $$\mathbb{R}^\mathbb{N}$$ and:
$$\mathcal{P}( \mathbb{N})$$
I thought about using these known facts:
If $A,B,C$ sets then:
1. $A^{B^ {~C}} \sim A^{B \times C}$
2. If $B$ and $C$ are disjoint then $A^B \times A^C \sim A^{B~ \cup ~C}$
But I am stuck on that proof.. I tried to set:
$A = \mathbb{R}$ , $B = \mathbb{N}$ , and $C = \{0\}$ and to use the known facts above while using $|A^B| = |A|^{|B|}$
I would appreciate your help, thank you!