Let $X$ be a compact space, $x_0\in X$, and define $$A=\{\{f_n\} ; f_n\in C(X), \sup_n\|f_n\|<\infty, and \{f_n(x_0)\} \text{ is a convergent sequence} \} $$ If $\|\{f_n\}\|$ is defined as $\sup_n\|f_n\|$ and the operations on $A$ are defined entrywise, show that $A$ is an abelian C*-algebras and find its maximal ideal space.
To find maximal ideal space of $A$, I think $A\{f_n\}$ is an ideal if and only if there is $n\in \Bbb N$, and $x\in X$ such that $f_n(x)=0$ for $\{f_n\} \in A$.
Also $I_x = \{ \{f_n\}\in A ; f_n(x)=0 ~~\text{for some n}\}$ for every $x\in X$ is a maximal ideal. I'm not sure that's correct or not. Please help me. Thanks.
Hints:
1) Solve the case when $X$ consists of one point.
2) $I_x$ is not an ideal.
3) The sets $I_{x,k}=\{(f_n)\mid f_k(x)=0\}$ are ideals. Consider also $I_{x,\infty}=\{(f_n)\mid \lim_kf_k(x)=0\}.$
4) Let $\widetilde{\mathbb N}=\mathbb N\cup\{\infty\}$ be the one-point compactification of $\mathbb N.$ Find a homomorphism from $A$ to $X\times \widetilde{\mathbb N}$ and show it is an isomorphism.