Solution
= Solution
Write $v_m=\operatorname{Var}(X_m)=\sigma_m^2$. Independence and $\mathbb E[X_{n+1}]=0$ give
$$
\begin{aligned}
\mathbb E[(S_{n+1}+c)^2\mid\mathcal F_n]
&=(S_n+c)^2
+2(S_n+c)\mathbb E[X_{n+1}]
+\mathbb E[X_{n+1}^2]\\
&=(S_n+c)^2+v_{n+1}\\
&\geq(S_n+c)^2.
\end{aligned}
$$
Thus
$$
\boxed{((S_n+c)^2)_{n\geq0}\text{ is a submartingale}.}
$$