I have been working on the following problem and was wondering if somebody can help me with one part of it.
Let $U$ and $V$ be open and connected sets in the complex plane $\mathbb{C}$, and $f:U\rightarrow\mathbb{C}$ be a holomorphic function with $f(U)\subseteq V$. Suppose that $f$ is a proper map from $U$ into $V$. Show that $f$ is surjective.
I have managed to show that necessarily $f(U)=V$ using the open mapping theorem and the fact $f$ is proper. Then my argument is that since $V$ is stated as just an open connected set such that $f(U)\subseteq V$ then we may take $V=\mathbb{C}$ because $\mathbb{C}$ quite obviously is open and connected.
Is this a sufficient argument? Or have I made a horrible assumption somewhere?