Let $(B_n)_n$ be a sequence of independent Bernoulli random variables with $P(B_n=1)=1/n, n \in \Bbb N$. How to show that there exists a subsequence $(B_ {n_k} )$ s.t. $\lim_{k \to \infty} B_{n_k}=0$? Is it possible to find a deterministic subsequence? Any help will be appreciated.
2026-03-28 17:41:54.1774719714
Convergence of a subsequence of Bernoulli random variables
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You can show that $B_n$ converges to 0 in probability, and then apply the standard result that convergence in probability implies the existence of a subsequence converging a.s.