There is a point p not in E s.t for any e>0

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The discussion centers on the proof that a proper subset E of R is not compact if there exists a point p not in E such that for any ε > 0, there exists a point q in E satisfying |p - q| < ε. The proof establishes that since p is a limit point of E and does not belong to E, E cannot be closed, leading to the conclusion that E is not compact according to the Heine-Borel theorem. The validity of the proof hinges on the understanding of limit points and the properties of closed sets.

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Let E be a proper subset of R. There is a point p not in E s.t for any e>0, there exists a point q in E s.t |p-q|<e. Prove that E is not compact.

Proof:

p is in R-E. For a e>0, p+e is in E. So R-E is closed on one side which implies E is open on one side. By using heine-borel thrm we can conclude that E is not compact.


Is this proof valid?
 
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Let N(p) be a neighbourhood of p. What is N(p) intersect E? Does it intersect E at any point other than p? Use this and what you know about closed sets/limit points.
 


From what you are saying, it seems that p is a boundary point of E. But p doesn't belong to E. Now, for a set to be compact in R, it must be what?
 

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