Let $S_n= X_1+\ldots X_n$ where $X_i$ are independent, with mean $0$ and finite variance (possibly not the same). Let $F_n$ be the natural filtration generated by the $X_i$.
a) Prove that for $c>0$ $(S_n+c)^2$ is a submartingale.
b) Using (a) and Doob's inequality show that for $x>0$ $$ \mathbb{P}\Bigl( \max_{1\leq m\leq n}S_m \geq x \Bigr) \leq \frac{\operatorname{var}(S_n)}{\operatorname{var}(S_n) +x^2}. $$
I have proved (a) (even though I don't see why $c>0$ is needed) butI don't know how to prove (b), especially because I don't see where $var(S_n)$ at the denominator comes from. This looks like Kolmogorov inequality but we don't have the absolute value for $S_n$ so we have a sharper bound.