Gamelin's Complex Analysis, Chapter 3, Section 2, Exercise 7 I am having  a very hard time with a problem from Gamelin's Complex Analysis.
Problem Statement and Hint: Show that if 0 and $\infty$ lie in different connected components of the
complement $C^*\backslash D$ of $D$ in the extended complex plane, then there
is a closed path $\gamma$ in $D$ such that $\int_{\gamma}d\theta\ne 0$. Hint. The hypothesis
means that there are  $\delta>0$ and a bounded subset $E$ of $C\backslash D$ such
that $0\in E$, and every point of $E$ has distance at least $5\delta$ from every
point of $C\backslash D$ not in $E$. Lay down a grid of squares in the plane
with side length $\delta$ and let $F$ be the union of the closed squares in
the grid that meet $E$ or that border on a square meeting $E$. Show
that $\partial F$ is a finite union of a closed paths in $D$, and that
$\int_{\partial f}d\theta=2\pi$
I've found a suggested proof online that says "Note by construction, $\partial F\subset D$. I've attached a picture which shows my grid of squares, set $D$ and set $E$ containing 0. I've outlined in green all of the squares that meet $E$ or that border on a square meeting $E$ (I think). The union of these closed squares in the grid is called F. 

My first question is, how is $\partial F$, the border of $F$, contained in $D$?
 A: One cannot simply choose $E$ as the component of $0$ in $D^c$, as  this component may fail to be at positive distance from the rest of $D^c$. The existence of $E$ with required properties follows from the fact that (due to $D^c$ being compact in the topology of the sphere) the connected components of $D^c$ coincide with quasicomponents. That is, if two points $a,b\in D^c$ lie in different connected components of $D^c$, there are two disjoint clopen sets that contain $a$ and $b$ respectively. A proof of this topological fact can be found here. But my interpretation of 

Hint. The hypothesis means that there are $\delta >0 $ and a bounded subset $E$ ... 

is that Gamelin allows the reader to take the existence of $E$ for granted. 

As for the question why $\partial F\subset D$, consider the following: if $z\in \partial F$, then at least one of the (closed) grid squares containing $z$  is not contained in $F$. Hence, this square is at a positive distance from $E$. Since this distance does not exceed $\delta\sqrt{2}$, the claim $z\in D$ follows from the properties of $E$.
A: It's not clear how $E$ is choosen. I would choose $E$ to be the connected component of $C^*\setminus D$ which contains $0$. In that case the statement seems ok. The set $F$ contains $E$ and its boundary is far from other connected components of $C^*\setminus D$ hence $\partial F$ must be contained in $D$.
