I am having troubles with one problem:

Let (M,d) be a complete metric space. For a compact set , and for a closed set such that

[

would that be true if K was closed but not compact?

OK, so far my idea is to show by contradiction

then we assume that the inf of the distance set is equal to zero, that implies x=y

now, given K is compact and a compact set is bounded, however x is not necessary element of F (since a closed set might not be bounded)

but then, am I proving with that the statement is false?