# 01a - α the sequence of functions deﬁned by f n t = n α...

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Modern Analysis 2 Test 1 Answer (1 or 2) and (3 or 4) and 5. 1. Let f : [ a,b ] R be continuous and nonnegative, with maximum value M . Prove that lim n ± Z b a f n ² 1 /n = M. 2. Let f : [ a,b ] R be continuous. Prove that there exists p between a and b such that Z b a f = ( b - a ) f ( p ) . 3. Let X be a metric space on which ( f n ) n> 0 is a sequence of continuous real-valued functions converging uniformly to f . Show that if ( a n ) n> 0 is a sequence in X converging to a then lim n →∞ f n ( a n ) = f ( a ) . 4. Determine for which real value(s) of
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Unformatted text preview: α the sequence of functions deﬁned by f n ( t ) = ( n α t ) 2 t 2 + (1-nt ) 2 converges uniformly on [0 , 1]. 5. For each n > 0 let f n : [0 , 1] → [0 , 1] be continuous; assume that each f n is diﬀerentiable on (0 , 1) with | f n | 6 M . Show that some subsequence of ( f n ) n> converges uniformly on [0 , 1]. 1...
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