Let $V$ be a Banach space and let $v$ and $v_1,v_2,\ldots$ belong to $V$.
We say that the series $\sum_{k=1}^{\infty}v_k$ converges unconditionally to $v$ if for each positive $\varepsilon$ there is a finite subset $F_\varepsilon$ of $\mathbb{N}$ such that $\lVert\sum_{k\in F}v_k -v \rVert<\varepsilon$ holds whenever $F$ is a finite subset of $\mathbb{N}$ that includes $F_{\varepsilon}$.
Now, let $V=l^{\infty}=\{\{x_n\}_{n=1}^{\infty}\subseteq\mathbb{R}\mid \lVert\{x_n\}_{n=1}^{\infty}\rVert_{\infty}<+\infty\}$. Show that there exists a sequence $v_1,v_2,\ldots$ in $V$ such that $\sum_{k=1}^{\infty}v_k$ converges unconditionally to some $v$, but does not converge absolutely.