Geometric sequences with accumulating ratios span c0 ($x_n=(t_n^j{)}_{j\ge 1},0<t_n<1,t_n\to 0$)

The closed linear span of these vectors is all of $c_0$. Otherwise the Hahn-Banach theorem gives a nonzero continuous functional annihilating them, represented by $a\in {\ell }_1$. At the first index $m$ where $a_m\ne 0$, divide ${\sum }_j{a}_j{t}_n^j=0$ by $t_n^m$. The remaining tail has magnitude at most $t_n{\sum }_{j>m}|a_j|\to 0$, forcing $a_m=0$, a contradiction. This is an elementary coefficient-isolation argument for density by annihilators.