How do you show this isomorphism?

Question

How do you show that $\mathbb{R} \backslash \mathbb{Q} \cong \mathbb{N}^{\mathbb{N}}$? What is a good starting point in showing this?

Answer 1

$|\mathbb{R}|>|\mathbb{Q}|$ $\implies$ $|\mathbb{R} \setminus\mathbb{Q}|=|\mathbb{R}|=|2^{\mathbb{N}}|=|\mathbb{N}^{\mathbb{N}}|.$