Convergence of a series obtained by selecting some terms of another convergent series

Let ∑ni=1an\sum^n_{i=1} a_n be convergent. Let II contain a subsequence of its indexes, that is, I⊂Z+I\subset \mathbb{Z}^+. Let each b=ai1+…+ainb = a_{i_1} + …+ a_{i_n} where each i∈Ii\in I. Now I know that each bb is a subsequence of a partial sum of {a}\{a\}, so each bb should converge since the series of ana_n is. But how about ∑∞i=1bi\sum_{i=1}^\infty b_i? Does this “partial sums of partial sums” converge?

Thanks

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(1) I suppose the series in the first line has upper limit ∞\;\infty\; . Is this correct? (2) It’s not necessarily true that each b\;b\; is a subsequence of the sequence of partial sums of the first series.
– DonAntonio
2 days ago

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