Uniform convergence rate of cumulative distribution functions Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables such that, for some random variable $X$, 
$$|X_n-X| = O(a_n),$$
almost surely, for some sequence of real numbers $\{a_n\}_{n=1}^{\infty}$ converging to zero. That is, $X_n$ converges to $X$ almost surely, at rate $a_n$. 
Of course, it follows that $X_n$ converges to $X$ in distribution. My question is whether the cumulative distribution function (cdf) $F_n$ of $X_n$ also converges uniformly to the cdf $F$ of $X$, at rate $a_n$. Specifically, does it hold that
$$\sup_{x \in \mathbb R} |F_n(x) - F(x)| = O(a_n)?$$
This seems true to me, but I do not have a proof. Thank you in advance for any references or suggestions on proving this.
Addendum: You may assume that the $F_n$ and $F$ are absolutely continuous, to avoid issues regarding discontinuity points in the last display above.
 A: The discontinuous case is false, and I think with a little hand-waving I'd say the continuous case is false also.  The issue is that $|X_n - X|$ is in "units" of $X$, whereas $|F_n(x) - F(x)|$ is in "units" of probability, so you cannot bound both with the same series.  
Anyway, counterexample: all the r.v.s are constants and $X = 1, X_n = 1 + b^n$ for some $0 < b < 1$.
Clearly $|X_n - X| = b^n$ surely (not just almost surely).
However, for any finite  $n$, there exists $x = 1 + b^n/2$ s.t. $F_n(x) = 0, F(x) = 1$ so the sup $= 1$.
You see what I mean when I wrote about the "different units"?  Another way to think about this is if you plot $F$ and all the $F_n$ on the same graph, with $x \in (-\infty, \infty)$ as the horizontal axis and $[0,1]$ as the vertical axis, then $|X_n - X|$ is a proxy for the horizontal separation but $|F_n - F|$ is the vertical separation.  My counterexample is just a series of step functions which get closer and closer horizontally but the vertical distance is always $1$ at the max point.
This counter-example is discontinuous but if you want, just use small uniform r.v. to change steps into ramps and you can prove things rigorously, e.g. $X_n = Unif(1, 1+b^n), F_n(1) = 0, F(1) = 1, \sup = 1$, etc.
