For 𝑆⊂𝑋 compact and a 𝑥∉𝑆 Show ∃𝜀>0, s.t. inf𝑦∈𝑆𝑑(𝑥,𝑦)≥𝜀>0 Then according to 𝑑(𝑥,𝑦) is continue, we can say 𝑓(𝑦)=−𝑑(𝑥,𝑦) is also continue. After that we can say (∃𝑦0)sup𝑦∈𝑆𝑓(𝑦)=𝑓(𝑦0) is true by Thm 28 OH. According to 𝑥∉𝑆, we can say 𝑓(𝑦0)≠0 is true. Hence we can take 𝜀=𝑑(𝑥,𝑦0) and inf𝑦∈𝑆𝑑(𝑥,𝑦)≥𝜀=𝑑(𝑥,𝑦0)>0 is true. □