This problem seems very hard to me, and any hints will be greatly appreciated.
Here is the question:
Let S be the space of bounded real sequences; for each $a\in S$ let $M_a$ be its least upper bound; let $X=\prod\limits_{a\in S}[-M_a,M_a]$; let $\alpha\colon\mathbb N\rightarrow X$ be the map defined by $(\alpha(n))_a=a(n)$; let $\beta\mathbb N$ be the closure of $\alpha(\mathbb N)$ in X.
(a) Prove that $\beta\mathbb N$ is compact and that $\alpha(\mathbb N)$ is a dense homeomorphic copy of $\mathbb N$ inside of $\beta\mathbb N$. Conclude that $\beta\mathbb N\neq \alpha(\mathbb N).$
(b) Prove for each $a\in S$ there exists a unique continuous function $\overline{a}\colon\beta\mathbb N\rightarrow [-M_a,M_a]$ such that $a=\overline{a}\circ\alpha$.
(c) Let $\mu\in\beta\mathbb N\smallsetminus\alpha(\mathbb N)$. Define $\lim_\mu(a)=\overline{a}(\mu)$. Prove that $\lim_\mu$ is always defined, if $a_k\rightarrow l$ then $\lim_\mu(a)=l$, and that $\lim_\mu$ is multiplicative and linear.
(d) Show that if $a,b\in S$ such that $b$ is obtained from $a$ by $b_k=a_{k+1}$, then it's not necessarily the case that $\lim_\mu(a)=\lim_\mu(b)$.
(e) Let $b=(0,1,0,1,\ldots)$, what can you say about $\lim_\mu(b)$?
Here is what I have done so far
(a) $\beta\mathbb N$ is a closed subspace of a compact space so it is compact. By definition $\alpha(\mathbb N)$ is dense in $\beta\mathbb N$. I have managed to prove that $\mathbb N\cong\alpha(\mathbb N)$ in the discrete topology. $\alpha$ is the homoemorphism, and $\alpha(\mathbb N)$ is not compact since $\mathbb N$ is not compact.
(b) I have managed to prove this part without too much difficulty. I took $\overline{a}$ to be the restriction of the projection map onto the $a'th$ component.
(c) I have managed to prove that $\lim_\mu$ agrees with the limits of convergent sequences. I'm not sure how to show that it is linear and multiplicative. If the sequences $a$ and $b$ are convergent then it just follows from the regular limit laws, but I don't know how to treat the case when $a$ or $b$ is not convergent.
(d) and (e) I think that these parts are related. I think that $\lim_\mu(b)$ can either be $0$ or $1$ depending on which $\mu$ we pick, and that the shifted sequence will have the other limit. I'm not sure if I'm thinking about this correctly though.