Show that every sequence in a Banach space such that $\{x_n\} \rightarrow 0$ has a subsequence $\{x_{n_p}\}$ such that $\sum_{p=1}^{\infty} x_{n_p} $ converges by showing $S_N = \sum_{p=1}^{N} x_{n_p}$ is a Cauchy sequence.
The convergent to the zero vector is what is confusing -- typically it is just to some other element? How do I incorporate this?
Hint: Since $\lim\limits_{n\to\infty}x_n=0$, then there exist subsequence $\{n_p:p\in\mathbb{N}\}\subset\mathbb{N}$ such that $\Vert x_{n_p}\Vert\leq 2^{-p}$.