 Solution

Since $X_n>0$, the pointwise inequalityholds: 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 toalmost surely, by part (ii). These conditional probabilities are bounded by one. The dominated convergence theorem and tower property of conditional expectation giveThis proves Positivity of the limiting Pólya urn proportion without needing its explicit beta distribution.