I am stuck in the middle of what you can see below: when the book says "we can repeat the same construction [...] horizontal strip arbitrary close to $z_2=0$":
- We define different sets $E_l$ on any open subset of $\Delta$: what does this mean? I tried to pick $A\subseteq\Delta$ and pose $E_l:=\{z_2\in A\;\;:\;\;\sup_{z_1\in\Delta}|f(z_1,z_2)|\le l\}$ but it doesn't work for what is to come: if we agree on calling $\Delta\times\Delta_{z_2^0,r}$ an horizontal strip, taking $A\subset\Delta$ such that $d(0,A)>0$ such a strip is not arbitrarily close to $z_2=0$. So I misunderstood.
- Next: what does $B\subset\Delta$ dense mean? Where is dense?
Many thanks!
Let $A$ be an open neighborhood of $z_2^0$ with radius $1/n$, where $z_2^0\in \Delta$ is arbitrary.
For large positive $l$, let us define $E_l := \{z_2 \in A : \sup_{z_1\in \Delta} |f(z_1,z_2)|\le l\}$. ...... Thus $E_l$ is closed. ...... $\bigcup_l E_l=A$. Thus, Baire's theorem ...... that $E_l$ has non-empty interior for large $l$. ......$f$ is holomorphic in $\Delta\times\stackrel{\circ}E_l$. (Notice that $\stackrel{\circ}E_l$ is contained in $|z_2-z_2^0|<1/n$.)
Let $B=\bigcup_{z_2^0\in \Delta, n\in \mathbb{N}} \stackrel{\circ}E_l$, then $f$ is holomorphic in $\Delta\times B$.
By its construction $B$ is dence in $\Delta$ since $z_2^0$ is arbitrary and $\stackrel{\circ}E_l$ is contained in a neighborhood of $z_2^0$ with radius $1/n$.
$\Delta\times B$ contains a horizontal strip $\Delta\times\Delta_{z_2^0,r}$ arbitrary close to $z_2=0$ since there is an $\stackrel{\circ}E_l(\supset\Delta_{z_2^0,r}) $ arbitrary close to $z_2=0$.