Let $X$ be a metric space. The following conditions are equivalent
(a) ${\rm asdim}\ X = n$
(b) $n$ is the smallest integer such that for every $R > 0$ there exists $n + 1 $ families $U_i\ i=0,1,2,\ldots,n$, and $S > 0$ such that each $U_i$ is $R$-disjoint, $S$-bounded and the families $U_i$ cover $X$
I cannot understand the proof of theorem.