Originally Posted by

**poorna** Regard Q, the set of all rational numbers, as a metric space, with

d(p,q)=|p-q|. Let E be the set of all P in Q such that 2<P^2<3. Show

that E is closed and bounded in Q, but that E is not compact.

Is E open in Q?

I could prove that E is closed, open and bounded. But will the following solution suffice to show that E is not compact?

I can find a sequence of points {Kn} = 1.7, 1.73, 1.732,.... in E. This is a sequence converging to 3^(1/2).

So {Kn} is an infinite subset of E which does not have a limit point in E.

So E is not compact.