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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 trivial collection $\mu$ of open subsets that 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

$$ \psi_{r} \subset \overline{\bigg(\bigcup_{\mu \in \psi} D(z,r).  \bigg)} \tag{1.2} $$

There exists a finite subset of $\Delta$ of $\mu$ such that

$$\psi_{r} \subset \bigcup_{\mu \in \Delta} \mu.  \tag{1.3}$$

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

$$F(z+N_{k}) = \prod_{k} (z + N_{k}). \tag{1.4}$$

Going back to $(1)$ our original claim now becomes,

$$\lim_{k \rightarrow \infty} \prod_{k} (z + N_{k}) \rightarrow h(z) \text{$\,$ for all z $\in$ $\psi$ }. \tag{1.5}$$

To finish our game, it's wise to note that since $\psi_{r}$ is compact, and since that $\psi_{r} \subset \Gamma$ where $\psi_{r}  \subset \Gamma$ is an closed disk, then $F(z+N_{k})  \to h$ uniformly on $\Gamma$.

Sat, 11 Aug 2018 01:13 GMT