 Solution

Use ${\mathcal{F}}_0=\{\varnothing ,\Omega \}$; the printed $\{\varnothing ,\mathcal{F}\}$ does not define the intended trivial sigma-algebra. With the simple symmetric random walk convention from part (a), $M_n$ is adapted and integrable because $|S_n|\le n$. Independence givesBy linearity, the pull-out property for the ${\mathcal{F}}_n$-measurable $S_n$, and preservation of measurable variables by conditional expectation,This proves the square-minus-time martingale of a simple symmetric random walk.