I was able to do (a) and (b). Just stuck on (c). I noticed that this is quiet similar to Theorem 1.3.7. But I'm not sure here if $\sigma(a_1, \cdots, a_n)$ is compact here. Also, let $\hat{a}: \Omega(A) \rightarrow \sigma(a_1, \cdots, a_n)$, where $\tau \rightarrow (\tau(a_1), \cdots, \tau(a_n))$. Can we still say that $\ Ran (\hat{a}) = \sigma(a_1, \cdots, a_n)$?
Thank you for your help!
The equality $\sigma(a_1,\ldots,a_n)=\operatorname{Ran}(\hat a)$ is the definition of $\sigma(a_1,\ldots,a_n)$.
As for compactness, $\sigma(a_1,\ldots,a_n)$ is the image of the compact set $\Omega(A)$. So all you need to show that the map $\tau\longmapsto (\tau(a_1),\ldots,\tau(a_n))$ is continuous.