This problem is a standard counterexample in measure theory in disguise:

Let and define

for , , and . In what sense does converge to zero?

If then

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.

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