elementary set theory - Injective and surjective functions in ℕ

Let $f:\mathbb N\to \mathbb N$ be an injective function.

Let $g:\mathbb N\to \mathbb N$ be a surjective function

prove that $f(n)≥g(n)$ for all $n\in \mathbb N$.

this exercise has been puzzling me a long time.
the most reasonable proof would by finding a contradiction, and by proving that surjective functions from $\mathbb N\to \mathbb N$ Would have to either be an identity function or a function that assigns different values to even and odd numbers ( I could be wrong)
I tried playing a bit with the properties of injective and surgective functions since an injective function in N would have to be strictly increasing.
p.s : I'm still a highschooler so I'm fairly ignorant.