The following question is from Ponnusamy and Silvermann Complex Variables with applications (Pg 395) and I need help in solving it.
Suppose that for each point in domain D there corresponds a neighbourhood in which a family F is equicontinuos. Show that F is equicontinuos on compact subsets of D.
I proved that family is locally uniformly bounded on D, which implies that F is uniformly bounded on each compact subsets of D. But F is not given to be a family such that each of it's member is analytic. If such has been given then I would have used: " If F is Locally uniformly bounded of analytic functions in a domain D, then F is equicontinuous on compact subsets of D".
Also, for part ii , I thought quite a lot but wasn't able to make any progress.
So, Please help me!
This is actually a general metric space result.
Claim: There exists $r\colon D\to\mathbb{R}$ continuous such that $B(z,r(z))\subseteq D$ for all $z$ and $\mathscr{F}\vert B(z,r(z))$ is equicontinuous.
Proof: Let $R\colon D\to\mathbb{R}\cup\{\infty\}$ be defined by $R(z)=\sup\{r\in\mathbb{R}:\mathscr{F}\vert B(z,R)\text{ is equicontinuous}\}$. Then the triangle inequality gives $R(z)\geq R(w)-\operatorname{dist}(z,w)$ for all $z,w\in D$ (with obvious meaning when $\infty$ are around) so $R$ is 1-Lipschitz. Now let $r=\min\{\frac12 R, 1\}$. $\checkmark$
For any compact $K\subseteq D$, we have $\inf_K r=r_K>0$. Cover $K$ by finitely many open balls of radius $\frac12r_K$, say centred at $z_1,\dots,z_n$. Given $\epsilon>0$, we find for each $i=1,\dots,n$ a $\delta_i>0$ by equicontinuity of $\mathscr{F}\vert B(z_i,r_K)$: $$ (\forall f\in\mathscr{F})(\forall z,w\in B(z_i,r_K))(\operatorname{dist}(z,w)<\delta_i\implies \operatorname{dist}(f(z),f(w))<\epsilon) $$ and we may assume $\delta_i\leq r_K$. Let $\delta=\frac12\min\{\delta_1,\dots,\delta_n\}$. Then for every $z\in K$, we have $B(z,\delta)\subseteq B(z_i,r_K)$ for some $i$. Hence for all $f\in\mathcal{F}$ and all $w\in B(z,\delta)$, we have $\operatorname{dist}(f(z),f(w))<\epsilon$ by the equicontinuity condition on $\mathscr{F}\vert B(z_i,r_K)$.