In Hahn and Epstein's book "Classical Complex Analysis" book they proved the following Open Mapping Theorem as a consequence of the Rouche's theorem; for $f(z)$ and $g(z)$ analytic on and inside a simple closed contour $C$ and $|f(z)| > |g(z)|$ for $z$ on $C$, then the functions $f(z)+g(z)$ and $f(z)$ have the same number of zeros inside $C$. Can someone explain me the very last sentence of the argument, "It follows that.....". Apologies just for the sake of that had to post the whole proof.
If $\beta \in D(f(a),\delta)$ then there exists at least one $z \in D(a,r)$ such that $f(z)-\beta=0$. Hence $\beta =f(z) \in f(D(a,r))$. This proves the first inclusion. For the second inclusion we have to assume that $D(a,r) \subset G$ which is possible because $G$ is open and $a \in G$. [ I think they forget to put this restriction on $r$].