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Prove that the Series of Functions Converges Uniformly

(See attached file for full problem description with equations)

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9.3-5
Let {f_n} (from n - 1 to infinity) be a sequence of functions on [a,b]
such that (f_n)'(x) exists for every x is an element of {a,b](n is an element of I) and
(1) {(f_n)(x_0)} (from n=1 to infinity) converges for some x_0 is an element of [a,b].
(2) {f_n'}(from n=1 to infinity) converges uniformly on [a,b].
Prove that {f_n}(from n = 1 to infinity) converges uniformly on [a,b].Show how this result may be used to weaken that hypothesis of 9.3I.

[Hint: For x is an element of [a,b] write

(f_n)(x) - (f_m)(x) = {[f_n (x) - f_n (x)] - [f_n (x_0) - f_m (x_0)]} + [f_n (x_0) - f_n (x_0)]

Apply 7.7A to obtain

(f_n)(x) - (f_m)(x) = {[f'_n (c) - f'_m (c)](x - x_0) + [f_n (x_0) - f_m (x_0)(x_0)].

Theorem 9.3I
If (for each n is an element of I) f'_n (x) exists for each x is an element of [a,b], if {f}(n=1 to infinity) is continuous on [a,b], if converges on [a,b] to f, and if [f'_n}(from n=1 to infinity) converges uniformly on [a,b] to g, then g(x) = f'(x) (a <= x <= b).
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That is, lim f'_n (x) = f' (x) (a <= x <= b)

We use the book Methods of Real Analysis by Richard Goldberg.

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Solution Summary

This solution is comprised of a detailed explanation to show how this result may be used to weaken that hypothesis of 9.3I.

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