Let $\{x_n\}_{n=1}^\infty$ be a countable subset of $\ell_1(\mathbb{N})$. Suppose that there exists a constant $C>0$ such that for all $y\in c_{00}(\mathbb{N})$, $$ \limsup_{n\to\infty}|\langle y,x_n\rangle|\leq C\|y\|_{\infty},\qquad(1) $$ Now could we prove (1) holds for all $y\in c_0(\mathbb{N})$?
For definition of $c_{00}(\mathbb{N})$, see $c_{00}(\mathbb{N})$. Note that $c_{00}(\mathbb{N})$ is dense in $c_0(\mathbb{N})$.
My attempt: For any $y\in c_0(\mathbb{N})$, there exists a coutable subset $\{y_k\}_{k=1}^\infty$ in $c_{00}(\mathbb{N})$ such that $\|y_k-y\|_\infty\to0$. Notice that for each $n$, we have $$ |\langle y,x_n\rangle|=\lim_{k\to\infty}|\langle y_k,x_n\rangle| $$ and hence $$ \limsup_{n\to\infty}|\langle y,x_n\rangle|=\limsup_{n\to\infty}\lim_{k\to\infty}|\langle y_k,x_n\rangle|\qquad(2) $$ If we are allowed to swap the limit of RHS in (2), then with (1) we have $$ \lim_{k\to\infty}\limsup_{n\to\infty}|\langle y_k,x_n\rangle|\leq C\lim_{k\to\infty}\|y_k\|_\infty=C\|y\|_\infty $$ which is the desired result. Now the concern is whether we can swap the limit in (2). Or any additional condition needed for $\{x_n\}_{n=1}^{\infty}$ to make the swap valid?
Update: Thanks for the counter example from Gred! If we impose the condition that for any $1<s\leq2$, $\{x_n\}_{n=1}^{\infty}$ is bounded in $\ell_s(\mathbb{N})$, will the result be true? In fact, the problem is from a paper and the authour aims to show $\{x_n\}_{n=1}^{\infty}$ is bounded in $\ell_1(\mathbb{N})$. So we can not assume $\{x_n\}_{n=1}^{\infty}$ is bounded in $\ell_1(\mathbb{N})$.
I doubt: Let $x_n=n^2 e_n$ $(n \in \mathbb{N})$. If $y \in c_{00}(\mathbb{N})$ then $\lim_{n \to \infty}|\langle y,x_n\rangle|=0$ (hence you can choose any $C>0$) but for $y=(1/k)$ we get $$ \lim_{n \to \infty}|\langle y,x_n\rangle| = \infty. $$ I think that if $(x_n)$ is bounded then the inequality extends to $c_{0}(\mathbb{N})$.