1. The problem statement, all variables and given/known data Notation: |a_n| is the absolute value of a_n. (s_n) signifies a sequences; s_n signifies the value of the sequence at a particular n. Problem: Let n, k be arbitrary elements in N. Let (a_n) be a sequence such that lim inf |a_n| = 0. Prove that there is a subsequence (an_k) such that the series, SIGMA an_k from k=1 to +infinity converges. 2. Relevant equations None. 3. The attempt at a solution I am just at a complete loss for how to solve this problem. Here's my outline: Let s_n = SIGMA |a_k| from k = 1 to n. Since for all n, |a_n| => 0, (s_n) is nondecreasing. Now if I can show that (s_n) is bounded, I'd know that (s_n) converges and thus would know there is a convergent subsequence (sn_j) where sn_j = SIGMA |an_k| from k=1 to j. Thus we'd say that SIGMA |an_k| from k=1 to j would be convergent. And thus we'd have that SIGMA an_k from k=1 to +infinity would be convergent because absolutely convergent series are convergent. But following this route means I have to prove that (s_n) is bounded and that is where I happen to be stuck. Can anyone help me out?