Show that the power series $\sum\_{n=0}^\infty \frac{z^n}{n!}$ converges uniformly for all $z$.

I know that the definition for uniform convergence is: We say ${\_}$ converges to uniformly on a subset of Ω iff for every $>0$ there is an $\_0∈ℕ$ so that if $>\_0$ then $|\_()−()|<$ for all $∈$.

BUT I'm not sure if I can use it the same for a power series or if there are different rules with that. Can I?
Also, what would the f(z) be for this? I get that $f\_n(z)=\frac{z^n}{n!}$, but how do I get $f(z)$?