Convergence of subsequence of partial sums implies full convergence? Define $S_n = \sum_{j=1}^n a_j$ a sum of real numers.
Suppose there is a subsequence such that $S_{n_j} \to S$ for some $S$, i.e., the infinite sum converges along this subsequence. Does this imply that $S_n$ converges too?
It seems like it should, since it's simply an index which is increasing but there may be a counterexample I can't think of..

No, if e.g. $a_j = (-1)^j$ then $S_{2k} = 0$ for all $k$ but the series itself does not converge.
