Is the set of all function with this condition... countable? Let $A=\{f|f:\Bbb{N} \to \{0,1\},lim _ {n\to\infty} f(n) =0\}$, now is A countable ?
I think it is countable because if we let $K_n=\{f|f:\Bbb{N} \to \{0,1\}, for \ every \ n \le m f(m) =0\}$ then every $K_n $ is not empty,countable and $A=\bigcup _{n\in \Bbb{N} } K_n$ ,now A is countable ?

If $f:\mathbb{N}\to \{0,1\}$ has the property $\lim_{n\to\infty} f(n)=0$, then this means that $f$ is eventually $0$. 
More precisely, there exists $N$ such that $f(n)=0$ for all $n\ge N$.
This means that $A$ is countable, because it is in bijection with the set of finite $\{0,1\}$ sequences.
