Let us suppose that $(X, \lVert \cdot \rVert)$ is a normed space over $\mathbb{R}$ which has a Schauder basis, that is, there is a sequence of vectors $(x_n)_n$ in $X$ such that for all $x \in X$ there is a unique sequence of coefficients $(a_n)_n$ such that \begin{equation} \lim_{n\to \infty} \left\lVert x - \sum_{k=1}^n a_k x_k \right\rVert = 0. \end{equation}
For $A \subseteq X$, let $\overline{\mathrm{co}}(A)$ denote the closed convex hull of $A$, that is, the intersection of all closed and convex sets which have $A$ as a subset.
Now, suppose that $x \in \overline{\mathrm{co}}\left( \left\{ \epsilon x_n \colon |\epsilon| = 1, n \in \mathbb{N} \right\} \right)$. Does it follow that there are sequences $(b_n)_n$ and $(c_n)_n$ of numbers in $[0,1]$ such that \begin{equation} \lim_{n\to \infty} \left\lVert x - \sum_{k=1}^n \left(b_k -c_k \right)x_k \right\rVert = 0 \end{equation} and \begin{equation} \sum_{k=1}^\infty b_k + \sum_{k=1}^\infty c_k = 1? \end{equation}