In the process to prove theorem 7.25 of the Principles of Mathematical Analysis, I can't exactly understand why
"Since K is compact, there are finitely many points $p_1,...,p_r$ in K such that to every $x \in K$ corresponds at least one $p_i$ with $d(x,p_i)<\delta$"
I understand the definition of compactness in chapther 2 of Rudin, and I can "heuristically" understand above statement. But I want to know the exact proof of above statement. Thank you!