statistics inequality

Question

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$

Answer 1

For every random variable $Y$, one has $$ c^2\cdot\mathbf 1_{|Y|\geqslant c}\leqslant Y^2. $$ Integrating, one gets $$ c^2\cdot P(|Y|\geqslant c)\leqslant E(Y^2). $$ Apply this to $$ Y=\text{____}. $$ And the thing even has a name...