Variance of the limit of randomly halved intervals

Start from $[0,1]$. Repeatedly replace the current interval by a random sub-interval of half its length whose left end is uniform over the allowed positions; for example the first step chooses $[x,x+0.5]$ with $x\sim U(0,0.5)$. The nested intervals shrink to a single point. What is that point's variance?

$1/36$. The limit point is the sum of independent left-end shifts $U_k\sim U(0,2^{-k})$, $k\ge1$. So its variance is $\sum_{k\ge1}\tfrac{4^{-k}}{12}=\tfrac{1}{12}\cdot\tfrac13=\tfrac1{36}$.