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.