Elias Stein : Real Analysis I cannot understand why this particular line in the text is true: 
" Moreover, there are $O(k^{d-1})$ cubes in  $\cal{Q}\ '$ "
For the text see 
http://books.google.com/books?id=2Sg3Vug65AsC&pg=PA12&lpg=PA12&dq=elias+stein+Moreover,+there+are+O%28k+d%C2%A11+%29+cubes+2+in+Q+0&source=bl&ots=DCMQaf06nz&sig=1V3GKoVbY9Xq8uOdJtD7kpYS9ug&hl=en&sa=X&ei=UQYgVO7yH5GayAS_1IL4Bw&ved=0CCAQ6AEwAA#v=onepage&q=elias%20stein%20Moreover%2C%20there%20are%20O%28k%20d%C2%A11%20%29%20cubes%202%20in%20Q%200&f=false

If one of the $d-1$ dimensional sides of the rectangle $R$ has dimensions
$$
l_1\times l_2\times l_3\times\cdots\times l_{d-1}
$$
then there are at most
$$
\begin{align}
&\lceil k l_1+1\rceil\cdot\lceil k l_2+1\rceil\cdot\lceil k l_3+1\rceil\cdots\lceil k l_{d-1}+1\rceil\\
&\le \lceil k\rceil^{d-1}\cdot\lceil l_1+1\rceil\cdot\lceil l_2+1\rceil\cdot\lceil l_3+1\rceil\cdots\lceil l_{d-1}+1\rceil\\
&=k^{d-1}\cdot\lceil l_1+1\rceil\cdot\lceil l_2+1\rceil\cdot\lceil l_3+1\rceil\cdots\lceil l_{d-1}+1\rceil
\end{align}
$$
cubes of side $\frac1k$ on that side of the rectangle, assuming $k$ is an integer.
This is the same for all $2d$ faces, and so there is some constant, depending on the dimensions of the $R$, so that the number of cubes is less than that constant times $k^{d-1}$; that is $O\left(k^{d-1}\right)$.
