How to put this into set notation? if you could please help me put this sentence into set notation, examine my (flawed?)attempts, and tell me where I went wrong with them?
So we call a function $f\colon \mathbb{N} \to \mathbb{N}$ zero almost everywhere iff $f(n)=0$ for all except a finite number of arguments. 
My question is "how do we represent the the set of functions that are zero almost everywhere?" 
Here is my attempt:
$E=\{f\colon \mathbb{N} \to \mathbb{N}\mid f(n)=0\}$
Thank you!

Your $E$ doesn't ever consider the "for all except a finite number of arguments". I think the most accessible answer (although perhaps not the one you are fishing for) is simply: 
$E=\{f:\mathbb{N} \to \mathbb{N} | \exists S \subset \mathbb{N}$ where $|S|<\infty$ with $f(n) \neq 0$ for $n \in S$ and $f(n)=0$ otherwise}
