Prove that the power set of $\aleph_0$ =$2^{\aleph_0}$

Question

I wanted to prove that the $\wp(A)$=$2^{|A|}$ but for $A$ is infinite. I have done this using mathematical induction. However, I was told that this only holds for a finite $A$ and that I must use another method if I was to show that the proof holds for an infinite case e.g. when $A=\aleph_0$

Thanks

Answer 1

Hint: Find a bijection between $\mathcal{P}(A)$ and the set of all functions $A \to \{ 0, 1 \}$.

Answer 2

HINT: Consider the following function $$ F:\mathcal{P}(A)\rightarrow 2^\mathbb{N}\\ F(X)=f_X $$ Where $f_X$ is the characteristic function.