Let, $\Omega = \mathbb{R}^{[0,1]}$, that is the set of all functions from $[0,1]$ to $\mathbb{R}$, equipped with the sigma field $\mathcal{A}$ which makes each one dimensional projection measurable. Show that $C[0,1] \notin \mathcal{A}$, where $C[0,1]$ is the set of all real valued continuous functions on $[0,1]$.
I encountered the above problem in an exercise sheet on Brownian Motion. The problem was followed by the following hint.
$\textbf{Hint} :$ If $\Sigma$ is a non-empty set and $\mathcal{F}$ is a collection of functions from $\Sigma$ to $\mathbb{R}$. Then show that, $\sigma(\mathcal{F}) = \bigcup_{\{f_1,f_2, \dots\}\subseteq \mathcal{F}} \sigma(f_1,f_2, \dots)$.
I got stuck while trying to prove the hint. Though, it is easy to see that $\bigcup_{\{f_1,f_2, \dots \}\subseteq \mathcal{F}} \sigma(f_1,f_2, \dots) \subseteq \sigma(\mathcal{F})$, I could not show that the other part, that is $\sigma(\mathcal{F}) \subseteq \bigcup_{\{f_1,f_2, \dots\}\subseteq \mathcal{F}} \sigma(f_1,f_2, \dots)$.
Any help regarding the hint or the main problem will be appreciated. Thank you.