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?