Let $X$ be a random variable with finite second moment in a probability space $(\Omega , \mathcal{F}, P)$ and let $\mathcal{G}$ be a sub - $\sigma$ algebra of $\mathcal{F}$ show that $$0 \leq Var(X | \mathcal{G}) \leq Var(X)$$
For the first one inequality i proved that $$Var(X | \mathcal{G}) = E[X^2| \mathcal{G}] - (E[X| \mathcal{G}])^2$$ and for the Jensen inequality the claim holds, but i have problems to show the second inequality $$Var(X | \mathcal{G}) \leq Var(X)$$ any hint or help i will be very grateful.
This is not true.
For a counterexample, suppose $X$ takes values $-1, 0, 1$ each with probability $1/3$ and let $\mathcal{G} = \sigma(\{X=0\})$.
Then on $\{X\neq 0\}$, $$\operatorname{Var}(X\mid \mathcal{G}) = 1 > 2/3 = \operatorname{Var}(X)$$