Proving Cluster Point Existence in E: Insights from Advanced Calculus

[ccox]

Prove that a is a cluster point of E if and only if the set (E intersection (a-r,a+r))\{a} is nonempty for each r > 0.

I have the forward implication done but the backwards implication is giving me some trouble. Could you explain it to me.

 

[StatusX]

What is your definition of "cluster point?"

 

[ccox]

A point a in the reals is called a cluster of E if (E intersection(a-r,a+r) contains infinitely many points for every r>0.

 

[Hurkyl]

Well, if you're having trouble proving something complicated, try proving something simple first.

"infinitely many points" is a lot... maybe you can prove that

if the set (E intersection (a-r,a+r))\{a} is nonempty for each r > 0​

then

(E intersection(a-r,a+r) contains one point for every r>0.​

Then what about two points? Three points?

 

[StatusX]

If there are only finitely many points in (a-r,a+r)-{a}, then there is some point closest to a, and so...