PL-OM 2023 Problem 6

For any real numbers $a$ and $b>0$ , define an extension of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$ . We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$ .

Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$ , there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$ , their extensions covers $A$ .