([KRD10, #3.1.D]) Let \(\{a_n\}\) be a sequence such that \(\lim_{n\to\infty}|a_n|=0\). Prove that there is a subsequence \(\{a_{n_k}\}\) of \(\{a_n\}\) such that \(\sum_{k=1}^{\infty}a_{n_k}\) converges.
Solution
1. Extract a rapidly decaying subsequence.Proof: since \(|a_n| \to 0\), for each \(k \ge 1\) we may choose an index \(n_k\) with \(n_1 < n_2 < \cdots\) and \(|a_{n_k}| \le 2^{-k}\) (indeed \(|a_n| < 2^{-k}\) for all sufficiently large \(n\), so pick any \(n_k\) larger than \(n_{k-1}\) with this property).
2. The subseries converges absolutely.Proof: \(\sum_{k=1}^\infty |a_{n_k}| \le \sum_{k=1}^\infty 2^{-k} = 1 < \infty\), so \(\sum_k a_{n_k}\) converges absolutely, hence converges.
3. Q.E.D.