This is a problem from my past Qual: "Let $\Omega$ be the unit disk and $f:\Omega\to \Omega$ be an analytic function. COnsider the sequence $\{f^n(z)\}$ where $f^n=f\circ\ldots\circ f$ ($n$--times). Prove that there is a subsequence that converges for every $z$."
I learn from Ahlfors' COmplex Analysis. And I don't think there is any tool in there that concern $f^n$. SO I don't even know where to approach. Since we are dealing with unit disk, Schwarz lemma comes to mind, but still there is barely any information to work with.