Proof verification (sets & logic)

The basic idea is good, but you rather want to take the union of $G_N$'s where $$G_N:=\{f:\Bbb N\to\Bbb Z:f(n)=0\text{ if } n>N\}$$

Your proof is correct. Here are some points of feedback.

• Your definition of $g$ depends on a specific $f$ and is actually just the same as that $f$. It does not really do anything in your proof, so just leave it out.
• It is not quite true that $G = \bigcup_{N=1}^\infty B_n$, because the maps in $B_n$ do not have domain $\mathbb{N}$. They can however easily be made into maps with domain $\mathbb{N}$ (which is clearly what you mean), you should say this (and how).
• A few small issues with notation, it is a bit sloppy. There is a $\}$ too many in the definition of $G$. The set $B_n$ depends on $n$ not on $N$ in its notation. That kind of thing. Not really a problem here, but in bigger proofs it will get confusing.