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Unformatted text preview: , 1] such that lim n →∞ Z 1 x k ϕ n ( x ) d x exists for k = 0 , 1 , 2 , ... Show that the limit lim n →∞ Z 1 f ( x ) ϕ n ( x ) d x exists for every continuous function f on [0 , 1]. 1 6. For n = 1 , 2 , 3 , .. , let f n ( x ) = ± 1 if x ∈ { 1 , 1 2 , ..., 1 n } otherwise. (a) Does the sequence { f n } ∞ n =1 converge uniformly on R ? Justify your answer. (b) Assume that α : R → R is an increasing continuous function, prove or disprove the following identity lim n →∞ Z 11 f n ( x ) d α ( x ) = Z 11 lim n →∞ f n ( x ) d α ( x ) . 2...
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 Spring '11
 NA
 Calculus, Topology, Derivative, Continuous function

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