Let $\{f_{n}(z)\}$ be a sequence of analytic functions in the upper half plane (in a Hilbert space $H$) and continuous on the real axis, such that
(1) $0<|f_{n}(x)|\leq 1$ for all $x\in \mathbb R$, for all $n$, and
(2) $\sup_{x\in\mathbb R}|f_{n}(x)|\to 0$ as $n\to \infty$. and
(3) $f_{n}\in L^{2}(\mathbb R)$ for all $n$, such that the sequence of $L^{2}$-norms is uniformly bounded.
If $\{a_{k}\}_{k\geq 1}$ is a sequence of real numbers with no limit point, how I can construct a sequence of analytic functions in the upper half plane and continuous on the real axis, say $\{g_{n}(z)\}$ in $H$, in terms of the sequence $\{f_{n}\}$, with the following properties:
(1') $0<\lim\limits_{n\to\infty}\big(\sup_{x\in\mathbb R}|g_{n}(x)|\big)\leq 1$, and
(2') $|g_{n}(a_{k})|\to 0$ as $n\to \infty$, for all $k$.
(3') $g_{n}\in L^{2}(\mathbb R)$ for all $n$.
i.e, the only difference in the new sequence is that the sup is not going to 0.