Let $S_n= X_1+\ldots+X_n$ where $X_i$ are independent, with mean $0$ and finite variance. Suppose $|X_m|\leq K$ $\forall m$. Use that $S_n^2-var(S_n)$ is a martingale to prove that for any $x>0$ $$ \mathbb{P} \left(\max_{1\leq m\leq n}|S_m|\leq x \right) \leq \frac{(x+K)^2}{var(S_n)} $$
I have shown that $S_n^2-var(S_n)$ is a martingale but I don't know how to proceed since usual inequalities work in the opposite direction