Let gn and g be uniformly bounded on [0, 1], meaning that there exists a singleM >0 satisfying |g(x)| ≤ M and |gn(x)| ≤ M for all n ∈ N and x ∈ [0, 1]. Assume gn → g pointwise on [0, 1] and uniformly on any set of the form [0, α], where 0 < α < 1. If all the functions are integrable, show that limn→∞ ) 10 gn = ) 1 0 g.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Let gn and g be uniformly bounded on [0, 1], meaning that there exists a singleM >0 satisfying |g(x)| ≤ M and |gn(x)| ≤ M for all n ∈ N and x ∈ [0, 1]. Assume gn → g pointwise on [0, 1] and uniformly on any set of the form [0, α], where 0 < α < 1. If all the functions are integrable, show that limn→∞ ) 1
0 gn = ) 1 0 g.

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