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Prove that converges uniformly on E

(See attached file for full problem description with equations)

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94.8
Let be a sequence of functions on E such that
where . Let be a nonincreasing sequence of nonnegative numbers that converges to 0.
Prove that converges uniformly on E
(Hint: See 3.8C)

Theorem 3,8C
Let be a sequence of real numbers whose partial sums form a bounded sequence, and let be a nonincreasing sequence of nonnegative numbers which converges to 0. Then converges.
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We are using the book of Methods of Real Analysis by Richard Goldberg

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This solution is comprised of a detailed explanation to prove that converges uniformly on E

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