Let $U_1,U_2,U_3,\dots$ be IID uniform on $[0,1]$. For each $n\geq 1$ let $$U_{1:n}<U_{2:n}<\dots<U_{n:n}$$ be the order statistic of $(U_1,\dots,U_n)$. Independent of the $U$ process there is a standard polya urn process $S=(S_n)_{n\geq 1}$ with distribution specified by $$(S_n)_{n\geq 1}~~\sim~~\big(1+\sum_{j=1}^{n-1} 1(V_j\leq V_0)\big)_{n\geq 1},$$ where $V_0,V_1,V_2,\dots$ are IID uniform on $[0,1]$. So it is clear that $\frac{1}{n}S_n$ converges alsmost surely towards some $V_0$ and $V_0$ is uniform on $[0,1]$.
Now let $B\subseteq [0,1]$ be any Borel set and $1_B(\cdot)$ the indicator function of $B$. Question: $$\text{Does $1_B(U_{S_n:n})$ converge almost surely as $n\rightarrow\infty$ towards $1_B(V_0)$?}$$
It is clear that $U_{S_n:n}$ tends to $V_0$ almost surely. Since the random variables in the question are $\{0,1\}$-valued, a.s. convergence means that they stay finally constant a.s..
Remark: This question is about some kind of '0-1'- representations of Borel sets. For each Borel set $B\subseteq [0,1]$ and $n\geq 1$ I'm interested in the $\{0,1\}$-string $$\big[1_B(U_{1:n}),1_B(U_{2:n}),\dots,1_B(U_{n:n})\big]~\in\{0,1\}^n.$$ For large $n$ these strings should 'look like' the set $B$ (modulo uniform distribution). What can one say about long substrings $11\dots 11$ for large $n$?