In class we found the asymptotic distribution of the sample median of iid uniform random variables. We did this by cases, considering odd-numbered and even-numbered samples separately. It occurred to me that one case would suffice if we could establish that adjacent order stats converge almost surely, and my intuition says that they do. (I'm guessing this is true whenever dealing with iid variables bounded in probability, but I'm specifically interested in uniform r.v.'s.)
I tried my hand at proving this, but so far I've come up short. I tried arguing as follows: Consider $X_i\stackrel{iid}\sim \text{Unif}(0,1)$, let $\epsilon>0$, then for any order stat $X_{(m)}$, a new draw has probability $\epsilon$ of falling in the region $(X_{(m)},X_{(m)}+\epsilon)$; and since $\sum_{k=0}^\infty (1-\epsilon)^k < \infty$, we have the desired convergence. But then I realized this fails since $X_{(m)}$ can move around, and so even if a new draw falls in the region $(X_{(m)},X_{(m)}+\epsilon)$, subsequent draws might pull $X_{(m)}$ and $X_{(m+1)}$ farther apart again.
