Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 124 / 2 / a

Solution

The event that infinitely many

A_{n}

occur is

lim sup_{n} A_{n} = ⋂_{m \geq 1} ⋃_{n \geq m} A_{n}

. For every

m

0 \leq P (lim sup_{n} A_{n}) \leq P (⋃_{n \geq m} A_{n}) \leq \sum_{n \geq m} P (A_{n}) .

The right-hand side tends to zero as the tail of a convergent series. Therefore the first Borel-Cantelli conclusion is

P (n lim sup A_{n}) = 0.

This proves the Borel-Cantelli first lemma without any independence assumption.

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