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Let A be a non-empty subset of X, ε be a positive real number and lastly, let $x_1, \cdots, x_m \in X$ such that $A \subseteq B(x_1; ε) ∪ · · · ∪ B(x_m; ε)$. Prove that there exists $a_1, . . . , a_p \in A$ such that $ A \subseteq B(a_1; 2ε) ∪ · · · ∪ B(a_p; 2ε)$ with $p ≤ m$.

I tried by taking an element in A and arguing that it must be in $B(x_k; ε)$. I thought of taking another arbitrary element in A and using the triangle inequality but that didn't really get me anywhere.

Evan Tseng
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mathfanatic
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  • Let $a_k\in A\cap B(x_k,\varepsilon)\neq\emptyset,$ Then the collection of such $a_k$ satisfies the conclusion, by the triangle inequality. Some of the intersections may be empty, that's why the quantity of $a_k$ can be less than $m.$ – Ryszard Szwarc Jul 12 '23 at 04:44

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