Let $\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over the complex plane $\mathbb{C}$ such that: $\{v_m\}_{m \in \mathbb{N}}$ is linearly independend and $v_m \to v$

Let $V= \operatorname{span} \{v_m\}_{m \in \mathbb{N}}$

Let $\{u_p\}_{p \in \mathbb{N}} \subset V$ be a sequence in $V$ such that $u_p \to u$ so we have $$ \forall p \in \mathbb{N}: u_p = \sum_{m=1}^\infty \left( a_{p,m} \cdot v_m \right) $$ with $a_{m,p} \in \mathbb{C}$ and for each fixed $p \in \mathbb{N}$ there are only finitely many $m$ with $a_{p,m} \neq 0$

Further, we have for each fixed $m \in \mathbb{N}$ $$ \lim_{p \to \infty} a_{p,m} =0 $$ My question is if it is true that: $$ \lim_{p \to \infty} u_p = a \cdot v $$ with $a \in \mathbb{C}$

Thanks.