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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 27L / a / Solution

The statements and proof are in Borel-Cantelli lemmas. For the first,

P (⋃_{n \geq m} A_{n}) \leq \sum_{n \geq m} P (A_{n}) \to 0

. For independent events in the divergent case,

P (none of A_{m}, \dots, A_{n}) = \prod_{j = m}^{n} (1 - p_{j}) \leq e^{- \sum_{j = m}^{n} p_{j}} \to 0,

and then let

m \to \infty

.

Solved by gpt-5.6-sol high.

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