$\sigma\mathcal C$ is the $\sigma$-algebra generated by $\mathcal C$. Show $\sigma\mathcal C\subset\sigma\mathcal D$ if $\mathcal C\subset\mathcal D$. If $\mathcal{C}$ and $\mathcal{D}$ are two collections of subsets of $E$. How do I prove the following:
$$\mathcal{C}\subset\mathcal{D}\implies\sigma\mathcal{C}\subset\sigma\mathcal{D}?$$

HINT: $\sigma\mathscr{C}$ is the intersection of all $\sigma$-algebras $\mathscr{A}\subseteq\wp(E)$ such that $\mathscr{C}\subseteq\mathscr{A}$; see here if this fact is not already familiar. If $\mathscr{D}\subseteq\mathscr{A}$, then certainly $\mathscr{C}\subseteq\mathscr{A}$.
