Let \(H(D)\) be the space of holomorphic functions on an open connected set \(D\subset \mathbb C\). Suppose \(f_n\) is a sequence in \(H(D)\) converging to a function \(f\) uniformly on compact subsets. Suppose also that \(f\) is a nonconstant function. Let \(a\in D\), \(\alpha =f(a)\).
Show there is a sequence \(a_n\in D\) which converges to \(a\), such that \(f_n(a_n)=\alpha \) for each \(n\).