Let $U$ and $V$ be non-empty open connected subsets of $\mathbb { C }$ and $f : U \longrightarrow V$ an analytic function. Suppose that for all compact subsets $K$ of $V , f ^ { - 1 } ( K )$ is compact. Show that $f ( U ) = V$.
Let $U$ and $V$ be non-empty open connected subsets of $\mathbb { C }$ and $f : U \longrightarrow V$ an analytic function. Suppose that for all compact subsets $K$ of $V , f ^ { - 1 } ( K )$ is compact. Show that $f ( U ) = V$.