# Thread: closed bounded and compact

1. ## closed bounded and compact

consider the set of rationals to be a metric space with metric, d(x,y)=|x-y|. Then let E be the set of all rationals q with 2<q^2<3. Prove that E is closed and bounded. But not compact. I can see that it is bounded. How do i prove the other two?

2. Originally Posted by poorna
consider the set of rationals to be a metric space with metric, d(x,y)=|x-y|. Then let E be the set of all rationals q with 2<q^2<3. Prove that E is closed and bounded. But not compact. I can see that it is bounded. How do i prove the other two?
If it were compact then it would be complete. Clearly there exists a rational sequence $\left\{q_n\right\}$ that is Cauchy in this interval and $q_n\to \sqrt{2}$

3. For example. Define $\gamma_{n+1}=\frac{1}{2}\left(\gamma_n+\frac{2}{\g amma_n}\right)$ with $\gamma_1=\frac{3}{2}$. It is easy to prove that this is monotonically decreasing and bounded below by $1$, and so we know it converges in $\mathbb{R}$, so it must be Cauchy there. And if it's Cauchy there it must be Cauchy here (since a Cauchy sequence is independent of the space it's embedded in). But, $\lim\text{ }\gamma_n=\sqrt{2}$.

4. Well not compact is the easy one to prove so I won't walk about it.

As for closed it is easier to show that the complement is open. WLOG x>0 and $x^2\ge 3$ then you can find $\epsilon$ such that $x^2\pm\epsilon\ge 3$. (note that if $x^2=3$ then x isn't rational)