I'm trying to do this problem from Gamelin's book:
Let $f_n(z)$ be a sequence of analytic functions on a domain (= open connected set) $D$ such that $f_n(D) \subset D,$ and suppose that $f_n$ converges to $f$ uniformly on each compact subset of $D$. Show that either $f(D)\subset D$, or $f(D)$ consists of a single point in $\partial D$.
Now, I can see that $f(D) \subset D\cup \partial D$. If $f$ is not constant, then by open mapping theorem, $f(D)$ is open, so no point in the boundary can be in the image of $f$, hence in this case $f(D) \subset D$. But if $f$ is constant, I can't figure out why the mage of $f$ must be a point in the boundary. Why can't the image be in $D$?
Thanks in advance.