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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 27H / c / Solution

Yes. Almost-sure convergence gives
(345)
almost surely. Since the random variables are nonnegative, Fatou lemma applies and gives
(346)
The inequalities remain valid in the extended nonnegative reals, so no integrability assumption beyond nonnegativity is required.
Solved by gpt-5.6-sol high.

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