diff --git a/arxiv_cbd.tex b/arxiv_cbd.tex
index 829c42bb3046c07dff64b6d893314226df67655d..10a965448a4ea86e902ac9f0630951735cce1faa 100644
--- a/arxiv_cbd.tex
+++ b/arxiv_cbd.tex
@@ -893,6 +893,7 @@ Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks
 To see that such bag is bounded consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any
 disk in $\beta(C)$ intersects $C'$ on an area at least $\pi\ell^2/16$. This implies that $|\beta(C)| \le 16t(2k+1)^2 / \pi$.
 
+Let us now give a detailed proof in a more general setting.
 For a measurable set $A\subseteq \mathbb{R}^d$, let $\vol(A)$ denote the Lebesgue measure of $A$.
 For two measurable subsets $A$ and $B$ of $\mathbb{R}^d$ and a positive integer $s$, we write $A\sqsubseteq_s B$
 if for every $x\in B$, there exists a translation $A'$ of $A$ such that $x\in A'$ and $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$.