If a unit ball is compact then why a ball of radius 5 has to be compact too? So if I use the definition of compactness that every open cover has a finite sub-cover, then as the unit ball is compact , there exists a finite subcover. But if I increase the radius of the ball, why does it still need to be compact. Intuitively speaking can't I just take very small sized and large number open sets in such a way that there is no finite sub-cover. I know that the answer to this question is no but I don't see why?
Can someone please throw some light?

Take your open cover of a ball of radius 5. Shrink it by a factor of 5. It's an open cover of the ball of radius 1.  Therefore it has a finite subcover.  Now reflate the subcover by a factor of 5.  Et Voila! It covers the ball of radius 5.
