Let $F$ be an entire function. We say that $a \in \mathbb{C} \cup \{\infty\}$ is an asymptotic value for $F$ if there exists a continuous curve going from a finite point to infinity such that $F$ tends to $a$ along that curve. Prove that for any non constant entire function $\infty$ is an asymptotic value.
I encountered this problem in an old qualifying exam. My thoughts on this are that if we assume that $\infty$ is not an asymptotic value, then for every continuous curve from a finite point to infinity $F$ tends to some finite value along that curve. One possibility is that $F$ tends to the same finite value in which case $F$ has a finite limit at $\infty$. Hence on the compact set $\mathbb{C}\cup\{\infty\}$, $F$ is bounded. Then we reach the contradiction that $F$ is bounded by Liouville's theorem. The other possibility is that $F$ does not have a limit at $\infty$ but takes on finite values in the neighborhood of $\infty$. In this case, we are probably getting a contradiction from Picard's Great Theorem.
Can someone please comment on this argument and point out possible wrong inferences or missing details?
