Here is the problem. Prove that

$\displaystyle ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\sim \mathcal{P}(\mathbb{Z^+})$

Where $\displaystyle ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})$ means the set of
functions from $\displaystyle \mathbb{Z^+}$ to $\displaystyle \mathcal{P}(\mathbb{Z^+})$.

So I have to come up with some kind of bijection between the two sets. Anything
which belongs to $\displaystyle \mathcal{P}(\mathbb{Z^+})$ is a subset of
$\displaystyle \mathbb{Z^+}$. I will need to associate this to some function from
$\displaystyle \mathbb{Z^+}$ to $\displaystyle \mathcal{P}(\mathbb{Z^+})$.

I am having difficulty coming up with such a bijection. Is it too difficult?. Or is there another approach to tackle this problem ?