(Reading Merging, Reputation, and Repeated Games with Incomplete Information by Sylvian Sorin and trying to understand a probabilistic argument):
Let $(\Omega,\mathcal{F},P)$ be a probability space equipped with a filtration $\{\mathcal{F}_n\}$ (increasing sequence of sub σ-fields of $\mathcal{F}$). We assume that each σ-field $\mathcal{F}_n$ is generated by a countable partition $\mathcal{F}_n^*$ and we denote by $F_n(\omega)$ the atom of $\mathcal{F}_n^*$ containing $\omega$.
Let $q=\{q_m\}$ be a martingale with values $[0,1]$.
Let $m,M>0$ such that $\mathbb{E}_P\left[(q_{m+1}-q_m)^2\right]\leq\frac{1}{M}$.
The article claims that this bound implies that $P\left(\omega;\mathbb{E}_P\left[(q_{m+1}-q_m)^2\mid F_{m-1}(\omega)\right]\geq\frac{1}{\sqrt{M}}\right)\leq\frac{1}{\sqrt{M}}$.
I'm struggling to find the inequality \ property used for this conclusion. Any help will be appreciated :)