This problem is a standard counterexample in measure theory in disguise:
Let and define
for , , and . In what sense does converge to zero?
Now, as , we have and so . Hence
so that converges to zero in probability.
It follows immediately that , as
and similarly , for as
However, does not converge almost surely to zero, as for we have
The “disguise” is that is really just the “typewriter sequence” from e.g. Terry Tao’s blog, with some minor tweaks that do not affect convergence.