= Solution
{c}
Since $X_n>0$, the pointwise inequality
$$
\mathbf1_{\{X_\infty=0\}}\leq\frac{(X_n-X_\infty)^2}{X_n^2}
$$
holds: on that event both sides equal one, and elsewhere the left side is zero. Take <conditional expectation> and pull out the positive $\mathcal F_n$-measurable denominator to obtain the requested bound. The supplied variance estimate improves it to
$$
0\leq\mathbb P(X_\infty=0\mid\mathcal F_n)
\leq\frac1{(m+n)X_n}=\frac1{B_n}\longrightarrow0
$$
almost surely, by part (ii). These conditional probabilities are bounded by one. The <dominated convergence theorem> and <tower property of conditional expectation> give
$$
\boxed{\mathbb P(X_\infty=0)=\lim_n\mathbb E[\mathbb P(X_\infty=0\mid\mathcal F_n)]
=\mathbb E[\lim_n\mathbb P(X_\infty=0\mid\mathcal F_n)]=0.}
$$
This proves <Positivity of the limiting Pólya urn proportion> without needing its explicit beta distribution.
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