Holomorphic function approaches infinity on boundary Let $f:U \to \mathbb{C}$ be a holomorphic function on an open, relatively compact domain $U$ of $\mathbb{C}$. If $f$ is unbounded on U, then evidently _there exists_ a sequence $(a_n)$ in $U$ which converges to some $a$ on the boundary of $U$, such that $|f(a_n)|$ diverges to infinity. But is this true for _every_ sequence in $U$ that converges to a point on the boundary?

No. Take$$\begin{array}{rccc}f\colon&D(0,1)&\longrightarrow&\mathbb C\\\&z&\mapsto&\frac1{1-z}\end{array}$$and $a_n=-1+\frac1n$ for each $n\in\mathbb N$.