I have studied the proof of Monotone Class Theorem from John B. Walsh's Knowing the Odds. I feel that a part of the proof is an overkill. I'm attaching the proof below.
The author has defined $\mathcal{G'}$ to be the smallest monotone class containing $\mathcal{F_0},$ but then defined the class $\mathcal{C_1}$ to show that $\mathcal{F_0} \subset \mathcal{C_1}=\mathcal{G'}$ and hence $\mathcal{G'}$ contains $\mathcal{F_0}$. If $\mathcal{G'}$ is defined to contain $\mathcal{F_0},$ why do we need to prove that again$?$
Any clarification on whether it is really an overkill or I'm just missing something would be much appreciated. Thank you.
The role of the sets $\mathcal C_1$ and $\mathcal C_2$ is not to show that $\mathcal G'$ contains $\mathcal F_0$; that is immediate from the definition of $\mathcal G'$, as you have noted. Their role is to show that $\mathcal G'$ is closed under finite intersections, which is the relevant remark in the second sentence you highlight.