 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-2/12f/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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 .

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