In the text "Complex Analysis" by Elias M. Stein and Rami Shakarchi is my proposed proof of $\text{Proposition (1)}$ sound ?
$\text{Proposition (1)}$
There exists and entire function $F$ with the following "universal" property: give any entire function $h$, there is an increasing sequence $\big\{N_{k} \big\}_{k=1}^{\infty}$ of positive integers, so that
$$\lim_{k \rightarrow \infty} F(z+N_{k}) = h(z) \tag{1.1}$$
uniformly on every compact subset of $\mathbb{C}$
$\text{Proof}$
To address $(1)$, first we let $\Gamma\subset \mathbb{C}$, such that there exists a compact set $\psi_{r}$ such that for every collection of open subsets denoted by $\mu$. There exists an $r > 0$ such that, for each $\mu \in \psi_{r}$ there is a closed disc $ \overline D(z,r)$ in $\Gamma$ such that
$$ \psi_{r} \subset \overline{\bigg(\bigcup_{\mu \in \psi} D(z,r) \bigg)}.\tag{1.2} $$
After constructing our spaces, a critical part of our game is to consider that $F(z+N_{k}): \psi \rightarrow \mathbb{C}$ and define that
$$F(z+N_{k}) = \prod_{k} (z + N_{k}). \tag{1.3}$$
Going back to $(1)$ our original claim now becomes,
$$\lim_{k \rightarrow \infty} \prod_{k} (z + N_{k}) \rightarrow h(z) \text{$\,$ for all z $\in$ $\psi$ }. \tag{1.4}$$
To finish our game, it's wise to note that since $\psi_{r}$ is compact, and since that $\psi_{r} \subset \Gamma$ where we have a closed disk contained, then $F(z+N_{k}) \to h$ uniformly on $\Gamma$ as $ k \rightarrow \infty$.