A  formal series in the Banach space $c_0$ Let $(e_n)$ be the standard unit vector basis for $c_0$, let $x_1=e_1$, $x_n=e_n-e_{n-1}$ when $n\gt1$.
Prove that the formal series $\sum x_n$ is not weakly subseries convergent.
 A: Here's an overkill argument: According to what is usually called the Orlicz-Pettis theorem, a series in a normed space is weakly subseries convergent (if and) only if it is subseries convergent in the norm. Your series clearly isn't norm-convergent.
More seriously: Note that if there is a weak limit of a sequence in $c_{0}$, it also is its weak$^{\ast}$-limit in $\ell^{\infty} = (c_{0})^{\ast\ast}$ (here $c_{0}$ is viewed as a subspace of $\ell^{\infty}$ via the canonical inclusion). Since the weak$^{\ast}$-topology is Hausdorff, it suffices to exhibit a subseries which is weak$^{\ast}$-convergent to some $s \in \ell^{\infty} \smallsetminus c_{0}$. The subseries $s_{k} = \sum_{n = 1}^{k} x_{2n} = (-1,1,-1,1,\cdots,-1,1,0,0,0,\cdots)$ weak$^{\ast}$-converges to $s = (-1,+1,\cdots)$: For all $(y_{n}) \in \ell^{1}$ and all $\varepsilon > 0$ there is $N$ such that $\sum_{n \geq N} |y_{n}| < \varepsilon$, therefore
\[
|\langle s - s_{k}, (y_{n}) \rangle_{\ell^{\infty}, \ell^{1}}|< \varepsilon
\quad \text{for $k \geq N$}
\]
by the Hölder inequality.
