Uniform integrability of reciprocal of random variables

Question

Let $\{X_n, n\geq 0 \}$ be a sequence of positive random variables that are uniformly integrable.

Assume that $\frac{1}{X_n}$ is integrable. Then, is it true that $\left\{\frac{1}{X_n}, n \geq 0\right\}$ is uniformly integrable?

Answer 1

No: consider the sequence $X_n(\omega):=1/n$ for each $n\in\mathbf N^*$. The sequence $\left(X_n\right)_{n\geqslant 1}$ is uniformly integrable but the sequence $\left(1/X_n\right)_{n\geqslant 1}$ is not even bounded in $\mathbb L^1$.