Let $X$ be an infinite set. I've been trying to prove that an injection from $X$ to $\mathbb{N}$ implies that $X$ is countable. I know this boils down to showing that an injection from $X$ to $\mathbb{N}$ implies the existence of a surjection from $X$ to $\mathbb{N}$. Or, equivalently, that a surjection from $\mathbb{N}$ to $X$ implies the existence of an injection from $\mathbb{N}$ to $X$.
Could someone give a proof of one of the above statements?