Advanced Calculus proof

  • Thread starter ccox
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  • #1
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
StatusX
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What is your definition of "cluster point?"
 
  • #3
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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.
 
  • #4
Hurkyl
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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?
 
  • #5
StatusX
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If there are only finitely many points in (a-r,a+r)-{a}, then there is some point closest to a, and so...
 

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