# Relative merits of proofs of the Borel–Cantelli lemma

by Michael Hardy   Last Updated July 08, 2019 20:19 PM

The first Borel–Cantelli lemma says:

Suppose $$\sum_{k=1}^\infty \Pr( A_k)<\infty.$$ Then $$\Pr(\text{infinitely many of } A_1,A_2,A_3,\ldots \text{ are true}) =0.$$

A frequently seen argument goes like this:

\begin{align} & \text{For each } n\in\{1,2,3,\ldots\}, \\[3pt] & \Pr(\text{infinitely many are true}) \\[6pt] \le {} & \Pr(A_n \text{ or } A_{n+1} \text{ or } A_{n+1} \text{ or } \cdots{}) \\ & \qquad \text{(inclusive “or'')} \\[6pt] \le {} & \sum_{k\,=\,n}^\infty \Pr(A_k) \to 0 \text{ as } n\to\infty \end{align} Now, where we have $$\Pr(\infty\text{-many}) \le \sum_{k\,=\,n}^\infty\cdots, \tag 1$$ the number on the left side of $$(1)$$ does not change as $$n\to\infty$$.

(That last line, set in bold is something one would usually omit if writing for mathematicians, and is an example of the sort of thing about which one should more often be explicit when writing for students.)

Another proof says the expected number of propositions $$A_1,A_2,A_3,\ldots$$ that are true is $$\sum_{k\,=\,1}^\infty \Pr(A_k) < \infty$$ and that could not be the case if $$\Pr(\infty\text{-many})>0.$$

What reasons are there to prefer one or the other of these arguments?

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