Denote by $(e_n)_{n=1}^\infty$ the canonical basis of $\ell_1$, and let $(w_n)_{n=1}^\infty$ be a sequence of positive real numbers. Define the vectors $(x_n)_{n=1}^\infty$ by setting $x_1=e_1$ and $$ x_n=w_ne_n-w_{n-1}e_{n-1},\;\;\;n\geqslant 2. $$ Is it true that $(x_n)_{n=1}^\infty$ is a Schauder basic sequence?
This is true in case $w_n\equiv c$ for some constant $c>0$. But, I would like to know whether it is true for an arbitrary sequence $(w_n)_{n=1}^\infty$ of positive scalars.
Thanks!