Indeed, there is no "closest" irrational number to p/q, because for any irrational number i , you can find another one j , which is closer to p/q, that means |j-p/q|<|i-p/q|.

This is because the irrational numbers are dense in R.

Hint:

First try to find an open covering of an open interval (a,b) , which has no finite subcovering.

Then pick out two irrational numbers a<b in [0,1] and consider the open covering (-1,a), (a,b), (b,2) of A.