Originally Posted by
poorna Let K be a suet of R1 consisting of 0 and 1/n for n = 1,2,3,...Prove that K is compact directly from the definition without uing the Heine Borel Property.
I am not sure if my solution to this is right. I was given a hint to use the Archimedian property of Reaal numbers. Can you tell me if my solution is right?
Let {Gi} be an open cover for K. Then one of these open sets should cover 1, call it G1. Since it is an open set in R1, it is an open interval. Then we can find a h>0, (where h< radius of G1) such that (1-h,1) is a subset of G1.
By the Archimedian property, there exist only finitely many n such that
n(1-h)>1
ie) only finitely many n such that , (1-h) > 1/n
So these finitely many n can be covered by finite open sets with together with G1 and the open set covering 0 form the finite subcover of {Gi}.
Am I right?