Weak convergence of a net in $c_0 (\mathbb{N})$.

Question

I try to prove the following result:

Let $((x_n^{(i)}))_{i \in I} \subset c_0 (\mathbb{N})$ be a net. Suppose $(x_n^{(i)}) \xrightarrow[]{w} (x_n)$. Then $x_n^{(i)} \to x_n$ for each $n \in \mathbb{N}$.

I am aware of the characterisation $(x_n^{(i)}) \xrightarrow[]{w} (x_n)$ iff $f (x_n^{(i)}) \to f(x_n)$ for all $f \in (c_0 (\mathbb{N}))^* \cong \ell^1 (\mathbb{N})$, but I am not sure how to proceed from here.

Any help is highly appreciated.

Answer 1

Hint: Note that for any $k \in \Bbb N$, the projection $f_k$ satisfying $$ f_k[(y^{(i)}] = y^{(i)}_k $$ is a bounded linear functional.