So as the title states I am trying to prove the following:
Show that if E ⊆ $\bigcap_{\alpha \in J}F_{\alpha}$, then $E^c \supseteq \bigcup_{\alpha \in J}F^c _{\alpha}$
I know that I should start by arguing: if $x \in \bigcup_{\alpha \in J}F^c _{\alpha}$ then $ x \notin F^{\alpha}$ for some $\alpha \in J$ However I have no idea where to go from here.
Any help is much appreciated, thanks.
$E \subseteq \bigcap_i F_i$ implies the complements reverse, so $$ \left(\bigcap_i F_i\right)^c \subseteq E^c $$ and now apply DeMorgan's Law.