Did I prove correctly that $f:\mathbb E\to \mathbb N;\quad f(x)=\frac12 x$ is surjective? Suppose we have two infinite sets, $\mathbb{N}$ (the set of natural numbers) and $\mathbb{E}$ (the set of even natural numbers). Give an example of a surjective function $\mathbb{E}\rightarrow\mathbb{N}$.
$f:\mathbb{E}\rightarrow\mathbb{N};\quad f(x)=\frac{1}{2}x$.
It is a surjection since $\frac{1}{2}x=y\implies\frac{1}{2}2x'=y\implies x'=y$ where $x=2x'$ and $x'\in\mathbb{N}$.
Is my argument correct? If so, is there anything I could improve?

Your argument is not entirely correct. In proofs like this one, phrases like "for all" and "there is" are very important. As it is currently written, I would not actually say that we have a solid proof.  
If you want to prove that $f$ is surjective, you must show/stress that for all $y\in\mathbb N$, there is an $x\in \mathbb E$, such that $f(x)=y$. Now argue that  $x=2y\,(\in\mathbb E)$ suffices and you're done.
