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Unformatted text preview: a B d ( a, ) Xf ( X ). Dene 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 ) : Induction on n 1. If n = 1 then clearly d ( a, x m ) , since x m f ( X ). Suppose d ( x n , x m ) for all m 6 = n . If m = 1 then d ( x n +1 , x 1 ) = d ( f ( x n ) , a ) . If m > 1 then d ( x n +1 , x m ) = d ( f ( x n ) , f ( x m1 )) = d ( x n , x m1 ) . For any x X the open ball B d ( x, / 3) has diameter less than or equal to 2 / 3, hence B d ( x, / 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

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