I'm trying to study for myself a little of Convex Geometry and I have a doubt with respect the proof of the Lemma 1.8.2 of the book Convex Bodies: The Brunn-Minkowski Theory. Before I presented the proof and my doubt, I will put the definitions used in the lemma below.
$\textbf{Definition:}$
(i) $\mathcal{C}^n := \{ A \subset \mathbb{R}^n \ ; \ A \neq \emptyset \ \text{and} \ A \ \text{is compact} \}$.
(ii) $B^n = \overline{B(0,1)} := \{ x \in \mathbb{R}^n \ ; \ d(x,0) \leq 1 \}$.
$\textbf{Lemma 1.8.2.}$ If $(K_i)_{i \in \mathbb{N}}$ is a decreasing sequence in $\mathcal{C}^n$, that is, if $K_{i+1} \subset K_i$ for $i \in \mathbb{N}$, then $\lim_{i \rightarrow \infty} K_i = \bigcap_{j=1}^{\infty} K_j$.
$\textbf{Proof:}$
The set $K := \bigcap_{j=1}^{\infty} K_j$ is compact and not empty. If the assertion is false, then $K_m \nsubseteq K + \varepsilon B^n$ for all $m \in \mathbb{N}$ with some fixed $\varepsilon > 0$. Let $A_m := K_m \backslash \ \text{int} (K + \varepsilon B^n)$; then $(A_m)_{m \in \mathbb{N}}$ is a decreasing sequence of nonempty, compact sets and hence has nonempty intersection $A$. Clearly, $A \cap K = \emptyset$, but $A_m \subset K_m$ implies $A \subset K$, a contradiction. $\square$
My doubt is why it's clear that $A \cap K = \emptyset$?
Thanks in advance!