TITLE: What is the point of nuking this mosquito in Real Analysis by Shakarchi and Stein?
QUESTION [7 upvotes]: I have tried to read volume 3 of Shakarchi and Stein more than a few times now and I keep getting stuck in chapter one in the same place.
After going through a bunch of basic concepts from analysis we finally come to a lemma, which says that if a given rectangle is the almost disjoint union of other rectangles, then the area of the given rectangle is equal to the sum of the area of the inner rectangles.  Ok duh. 
So at this point I assume we are going to have a rigorous proof which is going to explain why drawing lines of area 0 in a rectangle doesn't decrease the area of the rectangle.  Unfortunately the first sentence of the proof is "we consider the grid formed by extending the sides of all rectangles indefinitely." 
Is that supposed to be rigorous? I can think of three different ways to extend the sides of all the inner rectangles indefinitely, and I still can't figure out which one they mean because the picture is pretty strange, and so now I figure I am wrong about why the authors feel the need to prove such an obvious statement.
The next sentence is "This construction yields finitely many rectangles...and a partition of the integers between 1 and M".  I'm sorry, but is this a joke, or is this just a weak part of the book that I should skip over?
By the end of the proof I am left very confused as to what the point of this section is.  Can someone please explain what I am missing here?
Thanks!

REPLY [2 votes]: Unfortunately the first sentence of the proof is "we consider the grid formed by extending the sides of all rectangles indefinitely." 

Here is a construction of the grid in dimension $3$ (the general case being an obvious modification).
One starts from a rectangle $R=[x,y]\times[z,t]\times[u,v]$ which is the almost disjoint union of finitely many rectangles $R_k=[x_k,y_k]\times[z_k,t_k]\times[u_k,v_k]$. 
Consider all the coordinates $x_k$ and $y_k$ involved and order them as $x=a_0\lt a_1\lt\cdots\lt a_N=y$. Likewise, consider all the coordinates $z_k$ and $t_k$ involved and order them as $z=b_0\lt b_1\lt\cdots\lt b_P=t$ and consider all the coordinates $u_k$ and $v_k$ involved and order them as $u=c_0\lt c_1\lt\cdots\lt c_Q=v$.
Now, fix a bijection from $\mathfrak J=[NPQ]$ to $[N]\times[P]\times[Q]$, say $j\mapsto(n(j),p(j),q(j))$. For every $j$ in $\mathfrak J$, define 
$$
\tilde R_j=[a_{n(j)-1},a_{n(j)}]\times[b_{p(j)-1},b_{p(j)}]\times[c_{q(j)-1},c_{q(j)}].
$$
Then the rectangle $R$ is the almost disjoint union of the $M=NPQ$ rectangles $\tilde R_j$ and each original rectangle $R_k$ is the union of some $\tilde R_j$, that is, $R_k=\bigcup\limits_{j\in J(k)}\tilde R_j$, where the sets $J(k)$ are disjoint and their union is $\mathfrak J$.
Is all this preferable to Shakarchi and Stein's presentation? I am not sure.