Show that if $\displaystyle |X \times \mathbb{N}| = |X|$, then $\displaystyle |\{0,1\}^X| = |\mathbb{N}^X|$.

My attempt at the question is as follows:

Since $\displaystyle \{0,1\}^X \subseteq \mathbb{N}^X$, $\displaystyle |\{0,1\}^X| \leq |\mathbb{N}^X|$.

Since $\displaystyle |X \times \mathbb{N}| = |X|$, there exist 2 injective functions $\displaystyle f: X \times \mathbb{N} \longrightarrow X$ and $\displaystyle g: X \longrightarrow X \times \mathbb{N}$.

The problem here is what I should do to prove that $\displaystyle |\{0,1\}^X| \geq |\mathbb{N}^X|$.

Thank you in advance.