Limit of a sequence in $\ell^1$

Question

Let $\{s_j\}_{j \in \Bbb N} \in \ell^1$ be a linearly independent sequence of $\ell^1$ over $\Bbb C$ (finite sum $\sum_j c_j s_j=0 \Rightarrow$ scalars $c_j=0$)

Let $\{a_{m,j}\}_{m,j \in \Bbb N} \in \Bbb C$ such that for each $m$ then $a_{m,j}$ is with only a finite number of non-zero terms so $v_m=\sum_j a_{m,j}s_j \in V$

I would like to know if the following relation is true

$$\lim_{m \to \infty}a_{m,j}=0 \Rightarrow \lim_{m \to \infty} v_m=0$$

Thanks for any suggestion

Answer 1

As pointed out by b00n heT, the choice $a_{m,j}=1$ if $m=j$ and zero otherwise gives a counter-example. Indeed, in this case $v_m=s_m$ whose limit is not necessarily zero.