Here is the question: Let $(a_n)$ (for $n\in\mathbb{N}$) be a sequence of real number and let $S_k$ denote the partial sums of the series $\sum_{n=1}^\infty a_n$ Prove that if $\lim_{k\to \infty}$S_{2k}$=$\ \lim_{k\to \infty}$S_{2k+1}$=$L$, then $\sum_{n=1}^\infty a_n=L$
Here's my attempt:
$S_{2k}$ is the sequence of even partial sums such as: $(S_2, S_4, S_6, ...)$ and $S_{2k+1}$is the sequence of odd partial sums $(S_3, S_5, S_7,....)$ . There is a theorem in our book that states subsequences of a convergent sequence converge to the same limit as the original sequence. So can I say that since these are both converging subsequences of $S_k$ that converge to the same limit L, then $S_k$ converges to L?
Thanks.