The question is, Let $X$ be an $L^2(\mathbb{P})$ and $F_i$ be two sigma-fields such that $F_1 \subset F_2$. Let $Y_1 = E[X|F_1]$ and $Y_2=E[X|F_2]$. Show that $||X-Y_2||_{L^2} \leq ||X-Y_1||_{L^2}$.
By the definition, $Y_1 = E[X|F_1]$, then
i) $Y_1 \in F_1$ and ii) for all $A \in F_1$, $\int_{A} X dP = \int_{A} Y_1 dP$.
Since, $A \in F_1 \subset F_2, $ and , By the definition, $Y_2 = E[X|F_2]$, then
i) $Y_2 \in F_2$ and ii) for all $A \in F_2$, $\int_{A} X dP = \int_{A} Y_2 dP$.
I tried to use this definition to prove the inequality but did not reach to any conclusion. Is the tower property help here to prove this answer and how?