Compact Convergence of holomorphic functions

Question

A series of holomorphic functions $f_{0}, f_{1},...:D\rightarrow\mathbb{C}$ without any zeroes in the domain $D$ converges compactly against the holomorphic function $f:D\rightarrow\mathbb{C}$ also without any zeroes in $D$. Then the reciprocal of the series of functions converges compactly against $\frac{1}{f}$ in $D$. I have problems to verify this statement. By the definition of compact convergence, the series of functions has to converge uniformly on every compact subset of $D$. By this, there exists for everey $\varepsilon >0$ an $N\in\mathbb{N}$ such that $n\geq N$ so that for all $z$ from the compact subset: $|f_{n}(z)-f(z)|<\varepsilon$. I dont know how to apply this to the problem. Any hints?