= Solution
Choose $N$ with $T\leq N$. By part (d),
$$
M_T=M_0+\sum_{k=1}^NB_k\xi_k\mathbf1_{\{T\geq k\}}.
$$
Each $B_k\mathbf1_{\{T\geq k\}}$ is $\mathcal F_{k-1}$-measurable. Distinct summands are orthogonal: if $j<k$, conditioning on $\mathcal F_{k-1}$ makes every factor except $\xi_k$ known and $\mathbb E[\xi_k\mid\mathcal F_{k-1}]=0$. The cross terms with $M_0$ vanish in the same way, while $\xi_k^2=1$. Expanding the square therefore gives
$$
\begin{aligned}
\mathbb E[M_T^2]
&=M_0^2+\sum_{k=1}^N\mathbb E[B_k^2\mathbf1_{\{T\geq k\}}]\\
&=\boxed{M_0^2+\mathbb E\left[\sum_{k=1}^TB_k^2\right]}.
\end{aligned}
$$
This is the <stopped martingale isometry in a Rademacher filtration>.
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