theorem :: WEIERSTR:18

for M being non empty MetrSpace
for x being Point of M
for P being Subset of (TopSpaceMetr M) st P <> {} & P is compact holds
ex x2 being Point of (TopSpaceMetr M) st
( x2 in P & (dist x) . x2 = lower_bound ((dist x) .: P) ) by Th15, Th16;