Unformatted text preview: a ∈ B d ( a, ε ) ⊂ Xf ( X ). Deﬁne a sequence ( x n ) by x n = ± a, n = 1 f ( x n ) , n > 1 . If n 6 = m then d ( x n , x m ) ≥ ε : Induction on n ≥ 1. If n = 1 then clearly d ( a, x m ) ≥ ε , since x m ∈ f ( X ). Suppose d ( x n , x m ) ≥ ε for all m 6 = n . If m = 1 then d ( x n +1 , x 1 ) = d ( f ( x n ) , a ) ≥ ε . If m > 1 then d ( x n +1 , x m ) = d ( f ( x n ) , f ( x m1 )) = d ( x n , x m1 ) ≥ ε . For any x ∈ X the open ball B d ( x, ε/ 3) has diameter less than or equal to 2 ε/ 3, hence B d ( x, ε/ 3) cannot contain more than one point of A . It follows that x is not a limit point of A . ± 1...
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 Fall '08
 Brown
 Topology, Metric space, Topological space, bd, Morten Poulsen, open ball Bd

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