= Solution
Two applications of the <Cauchy-Schwarz inequality> give
$$
\left|\frac1n\sum_iF_iG_i\varepsilon_i\xi_i\right|
\leq
\left(\frac1n\sum_iF_i^2G_i^2\right)^{1/2}
\left(\frac1n\sum_i\varepsilon_i^2\xi_i^2\right)^{1/2}
\xrightarrow{p}0
$$
and
$$
\frac1n\sum_i|\varepsilon_iF_i|G_i^2
\leq
\left(\frac1n\sum_iF_i^2G_i^2\right)^{1/2}
\left(\frac1n\sum_i\varepsilon_i^2G_i^2\right)^{1/2}
\xrightarrow{p}0.
$$
Here the empirical residual second moment is again $O_p(1)$ by the <weak law of large numbers>, while the assumed product error and the conclusion of part b are $o_p(1)$.
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