Online I found the following statement that for $X$ an infinite set, if there is a bijection from $X$ to $\mathbb{N}$ this is equivalent to there is an injection from $X$ to $\mathbb{N}$. How would I prove this?
Attempt: The fact that if there is a bijection it is also an injection is true by definition of bijection. Beyond that, I don't know how to demonstrate that an injection is equivalent to a bijection for an infinite set $X$ to $\mathbb{N}$.
We start off by observing that $X$ can not be uncountable as $card(X) \le card(\mathbb{N})$ due to the existence of an injection $f: X \to \mathbb{N}$. However, since $X$ is infinite, it is countable. Hence, $X$ is in bijection with $\mathbb{N}$.