Uniformly integrable sequence tending to 0 a.s. but with $\mathbb{E}(\sup_n|X_n|) = \infty$

Question

I am trying to find a uniformly integrable sequence of random variables $(X_n: n \in \mathbb{N})$ such that both $X_n \to 0$ almost surely and $\mathbb{E}\left(\sup_n|X_n|\right) = \infty$. I think this is probably a well known counterintuitive result in the theory, but I haven't come across it before, so any help would be great.