This is more of a "How to write" question than a "help me solve" one, sorry if these are unaccepted/closed, let me know and I won't open anymore like this.
I need to prove that $A:=\{x\in \mathbb{N}|$ exists $n\in\mathbb{N}$ such that $x=n^2 \}$ is countable. This obviously requires me to prove that exists a bijective function from N to it. Which sounds very simple, but I don't really know how to write it.
Edit: Well, this is what I had written before asking:
Proof. We'll notice that this set is equal to $\left\{ n^{2}|n\in\mathbb{N}\right\}$ , since for each n we can find $n^{2}$ and say it's equal to $x$ .
In order to prove that a set is countable we need to find a bijective function from $\mathbb{N}$ to it. We'll look at:
$f:\mathbb{N}\to \mathbb{N}^{2},f\left(n\right)=n^{2}$
Since multiplication is well defined we can say this function is injective. Now how do I explain surjective?