= Solution
Hilbert-Schmidt convergence implies
$$
\lVert L_i^{(p)}\rVert_{\rm HS}^2
\longrightarrow\lVert L_i\rVert_{\rm HS}^2.
$$
The <Schatten norm Hölder inequality> gives
$$
\begin{aligned}
\lVert(L_2^{(p)})^*L_1^{(p)}-L_2^*L_1\rVert_1
&\leq\lVert L_2^{(p)}-L_2\rVert_{\rm HS}
\lVert L_1^{(p)}\rVert_{\rm HS}\\
&\quad+\lVert L_2\rVert_{\rm HS}
\lVert L_1^{(p)}-L_1\rVert_{\rm HS}
\longrightarrow0.
\end{aligned}
$$
The sum of singular values is the trace norm, and the reverse triangle inequality gives
$$
\left|
\lVert(L_2^{(p)})^*L_1^{(p)}\rVert_1
-\lVert L_2^*L_1\rVert_1
\right|\longrightarrow0.
$$
Substitution into part b proves
$$
d_P(C_1^{(p)},C_2^{(p)})^2
\longrightarrow d_P(C_1,C_2)^2.
$$
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