We are in , where is the Euclidean metric. Let our region be the open ball centered at the point with radius 1, denoted by .
My attempt: Let where .
It's obvious to me (not sure if I must prove) that is an open cover for since
Suppose there exists a finite subcover of , say . I must find an element in such that to reach a contradiction, right?
Since is a finite subcover we have that . So every point in is also in . Consider . It's obvious that , so
since the radius is bigger, but
since the radius is still less than 1. We have reached a contradiction.
Is this a correct?