= Geometric sequences with accumulating ratios span c0
{title2=$x_n=(t_n^j)_{j\geq1},\quad0<t_n<1,\ t_n\to0$}
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 $a_m\ne0$, divide $\sum_ja_jt_n^j=0$ by $t_n^m$. The remaining tail has magnitude at most $t_n\sum_{j>m}|a_j|\to0$, forcing $a_m=0$, a contradiction. This is an elementary coefficient-isolation argument for <density by annihilators>.
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