The axiom of countable choice states that for every countable collection $\{S_n\}_{n \in \mathbb{N}}$ of non empty sets there is a function $f : \mathbb{N} \to \bigcup_{n \in \mathbb{N}} S_n$ such that $f(n) \in S_n$ for every $n \in \mathbb{N}$.
For every set $X$ I can apply the axiom of countable choice to prove that there exists a function $f : \mathbb{N} \to X$ simply considering the case in which $S_n = X$ for every $n \in \mathbb{N}$.
Now, if $X$ is an infinite set, how can I use the axiom of countable choice to prove that there exists an injective function $f : \mathbb{N} \to X$?