Okay, it seems I wrote a confusing formulation of my question:
$\textbf{(1)}$ I'm dealing with the following problem: prove that if $ \left(\{a_n\}_{n=1}^{\infty}\right)_k \in \ell_p$, then
$$\left(\{a_n\}_{n=1}^{\infty}\right)_k \text{ converges weakly to }\left(\{a_n\}_{n=1}^{\infty}\right)_0 \in \ell_p\iff $$
$$ \iff \left(\{a_n\}_{n=1}^{\infty}\right)_k \text{ is bounded in }\ell_p \text{ and } \left(\{a_n\}_{n=1}^{\infty}\right)_k \text{ converges coordinatewise} $$
$\textbf{(2)}$ I have managed to prove a more general theorem: a sequence $ a_n $ in a normed space converges weakly to $ a $ if and only if it is bounded and $ \varphi(a_n) \rightarrow \varphi(a) $ for all $ \varphi \in D$ for some $ D $ such that $ \overline{\text{span}(D)} = X^* $
Now my problem lies in the following:
$ \textbf{(Question)} $ Is it true that $ \ell_p^* = \overline{\text{span}(\pi_1, \pi_2, \dots)} $, where $ \pi_k $ is a projection of the $ k $-th coordinate? How can it be proved or disproved?
It's easy to prove that for all $ \psi\in \ell_p^* $ there exists a sequence $ c_1\pi_{k_1} + \dots + c_n\pi_{k_n}$ converging to $ \psi $ for all fixed points in $ \ell_p $, but my aim is to prove that:
$$ \forall _{\varepsilon>0, \psi \in \ell_p^*} ~\exists_{c_1, \dots, c_k, n_1, \dots, n_k} $$
$$\text{such that}$$
$$|| \psi - \sum\limits_{i=1}^k c_k \pi_{n_k}|| < \varepsilon$$