Question about Titchmarsh's proof of the Vitali Convergence Theorem Consider the following version of the Vitali Convergence Theorem presented in Titchmarsh's Theory of Functions:

Let $f_{n}(z)$ be a sequence of functions, each regular in a region $D$; let $|f_{n}(z)| \leq M$ for every $n$ and $z$ in $D$; and let $f_{n}(z)$ tend to a limit, as $n \rightarrow \infty$, at a set of points having a limit point inside $D$. Then $f_{n}(z)$ tends uniformly to a limit in any region bounded by a contour interior to $D$, the limit being, therefore, an analytic function of $z$.

Titchmarsh starts the proof off as follows:

It is sufficient to consider the case where $D$ is a circle, and the limit point is its centre. For then, returning to the general case, we can prove uniform convergence in a circle with centre the limit point interior to $D$. Then we can repeat the process with any point of this circle; and so, by the method used in analytic continuation, extend the domain of uniform convergence to any region bounded by a contour interior to $D$.
We may also take the limit point as the origin...{Proof Continues}

The part I'm confused with is the last sentence "Then we can repeat the process with any point of this circle..." Suppose I've proven the result in the case when $D$ is a circle with centre at the limit point. Now I want to prove the general case. Denote the limit point by $p$. Then we have the result in the largest circle contained in $D$ centred at $p$, but how can we extend this?


Then we have the result in the largest circle contained in $D$ centred at $p$, but how can we extend this?

Any point of the disk centered at $p$ can be taken as a new point $p$. So, you get the convergence result for disks centered at any point of the first disk, then for disks centered at any point of any of the 2nd-generation disks, etc. Consider a closed disk $B$ contained in the domain (let's call the domain $\Omega$ to distinguish it from disks). I claim that convergence is uniform on $B$. Let $q$ be the center of $B$ and let $\gamma$ be a piecewise linear curve connecting $p$ to $q$ within $\Omega$. The distance from $\gamma$ to $\partial \Omega$ is a positive number, call it $r$. For every point $z$ on $\gamma$ the open disk of radius $r$ centered at $z$ is contained in $\Omega$. Since the length of $\gamma$ is finite, you can go from $p$ to $q$ in finitely many steps of size less than $r$. 
Now that you have uniform convergence on every closed disk contained in $\Omega$, a compactness argument (selecting a finite cover by such disks) gives uniform convergence on every compact subset of $\Omega$. 

Titchmarsh's book is pretty old: a modern text would state the result in terms of compact subsets of   $D$ instead of "any region bounded by a contour interior to $D$". 
