In this question, what is the purpose of the summation?
If the limit of the sequence is zero, the corresponding series is convergent. Does the desired conclusion not then follow from BC1?
2026-03-25 23:38:05.1774481885
Question on a variation of Borel Cantelli Lemma
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It is not true that if $a_n\to0$ as $n\to\infty$, then $\displaystyle\sum_{n=1}^\infty a_n$ converges.
The most widely known counterexample is that $\displaystyle\sum_{n=1}^\infty \frac 1 n=\infty$.