What exactly is the relationship between weak* convergence and sequential convergence?

Question

I am studying Banach algebras, and the theory frequently makes use of the weak* topology which I am not very familiar with. As I understand it, the weak* topology is defined as follows: let $X$ a normed vector space (or a topological vector space), and let $X^*$ be the dual space to $X$. Define the weak* topology on $X^*$ to be the coarsest topology such that the maps $T_x:X^*\to \mathbb{C}$ defined by $\phi\mapsto\phi(x)$ are continuous for all $x\in X$. On the Wikipedia page, convergence in the weak* topology is defined in terms of nets or sequences, saying that $\phi_\lambda\to\phi$ if $\phi_\lambda(x)\to\phi(x)$ for all $x\in X$ where $\lambda$ is a net, or in the case of sequences, $\phi_n\to\phi$ if $\phi_n(x)\to\phi(x)$ as $n\to\infty$ for all $x\in X$.

Now, on Wikipedia, they make the statement that "Weak-* convergence is sometimes called the simple convergence or the pointwise convergence. Indeed, it coincides with the pointwise convergence of linear functionals."

I have a couple questions about this:

1. I know nothing about nets, so I am wondering if I can say the following: if $\phi_\lambda\to\phi$ in the weak* topology with $\lambda$ a net, may I assume that there is a sequence so that $\phi_n\to\phi$? For example, if I wanted to prove that the set of characters on a unital abelian Banach algebra are closed, here is how I might try to prove it. Suppose $\phi_n$ is a sequence of characters of the unital abelian Banach algebra $X$ converging to $\phi\in X^*$, then $X$ being unital implies that $\phi$ is not the $0$ functional. Then $\phi_n(ab)=\phi_n(a)\phi_n(b)$ so taking limits on both sides yields $\phi(ab)=\phi(a)\phi(b)$. Thus the limit is nonzero and multiplicative, so is a character on $X$. Is this correct?

2. If the above is true, i.e. that you can see every weak* limit as a sequential limit in the above sense, then what information are nets providing about the space? Is there any reason to consider nets when using the weak* topology?

Apologies if the questions are rather basic, I would just like to know how much I need to know about nets before proceeding on with my study of Banach and C$^*$ algebras.