Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq 0,\dots S_n\geq 0) \leq \frac{C}{\sqrt{n}}.$$
I actually have no idea in how to approach this. In the case of simple random walk, we can use some reflection principle, but what about here? I don't have any clues.