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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 12F / a / Solution

By Tonelli theorem for the nonnegative indicators,

E [N] = E [\sum_{n \geq 1} 1_{A_{n}}] = \sum_{n \geq 1} P (A_{n}) .

If this is finite, then

N

is finite almost surely: alternatively, Markov inequality gives

P (N \geq m) \leq \frac{E [ N ]}{m} ⟶ 0.

Since

{N = \infty} \subseteq {N \geq m}

for every

m

,

P (N = \infty) = 0 .

This is the first Borel-Cantelli lemma.

Solved by gpt-5.6-sol high.

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