= Positivity of the limiting Pólya urn proportion
{c}
{title2=$B_0,R_0>0\implies\mathbb P(0<X_\infty<1)=1$}
Both colour counts tend to infinity almost surely. For example, $X_n\geq B_0/(m+n)$, and independent events $U_{n+1}\leq B_0/[2(n+1)]$ occur infinitely often by the <Borel-Cantelli second lemma>, forcing infinitely many blue additions.
The martingale increment formula gives $\mathbb E[(X_{k+1}-X_k)^2\mid\mathcal F_k]\leq X_k/(m+k+1)^2$. Conditional orthogonality of later increments and $\mathbb E[X_k\mid\mathcal F_n]=X_n$ imply
$$
\mathbb E[(X_\infty-X_n)^2\mid\mathcal F_n]\leq X_n\sum_{j=m+n+1}^\infty j^{-2}\leq\frac{X_n}{m+n}.
$$
On $X_\infty=0$ the squared difference is $X_n^2$, so its conditional probability is at most $1/[(m+n)X_n]=1/B_n\to0$. Dominated convergence proves zero probability of that endpoint. The symmetric argument for red excludes $X_\infty=1$. No explicit beta law is needed for this proof.
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