Is this set uncountable or countable?

I have to prove whether or not this set is countable: the functions from $\mathbb Z$ to $\mathbb R$ such that $f(n)=0$ except for a finite number of $n \in \mathbb Z$.
I think this set is uncountable. So, if $f$ belongs to this set, there is an $n_0$ such that $f(n)=0$ $\forall$ $n>n_0$. I tried to prove that the set of functions for which this is true for a certain $n_0$ is uncountable but I'm having a hard time finding a function to do so... Thanks

HINT Find an injection from $\Bbb R$ into the set of functions by considering only functions which satisfy $g(n)=0$ for all $n\neq0$.
• (1) You can use a similar idea to define a surjection from the set of functions onto $\Bbb R$, that alone should suffice to prove the set is uncountable; (2) the injection is in the other direction, note that functions that have at most one nonzero element in their range at the point $0$ must be equal iff the agree on that element. – Asaf Karagila Apr 3 '14 at 1:09
Explicitly, you could note that $\forall r\in\mathbb{R}$, the function that is $r$ at zero and 0 everywhere else is in the set. There are obviously uncountably many such functions.