= Solution
{c}
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 $|S_n|\leq n$. Independence gives
$$
\mathbb E(X_{n+1}\mid\mathcal F_n)=0,\qquad
\mathbb E(X_{n+1}^2\mid\mathcal F_n)=1.
$$
By linearity, the pull-out property for the $\mathcal F_n$-measurable $S_n$, and preservation of measurable variables by <conditional expectation>,
$$
\begin{aligned}
\mathbb E(M_{n+1}\mid\mathcal F_n)
&=S_n^2+2S_n\mathbb E(X_{n+1}\mid\mathcal F_n)
+\mathbb E(X_{n+1}^2\mid\mathcal F_n)-n-1\\
&=S_n^2-n=M_n.
\end{aligned}
$$
This proves the <square-minus-time martingale of a simple symmetric random walk>.
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