Extension of Pratt's Lemma

Pratt's Lemma is : $\xi,\eta,\zeta$ and $\xi_n,\eta_n,\zeta_n$ such that: $$ \xi_n\rightarrow \xi, \eta_n \rightarrow \eta, \zeta_n\rightarrow \zeta, \text{convergence in probability} $$ and $\eta_n\le \xi_n\le \zeta_n$, $E\zeta_n\rightarrow E\zeta,E\eta_n\rightarrow E\eta$, and $E\zeta,E\eta,E\xi$ are finite, prove :

If $\eta_n\le 0\le \zeta_n$, then $E|\xi_n-\xi|\rightarrow 0$.

I know how to prove $E\xi_n\rightarrow E\xi$, but $E|\xi_n-\xi|\rightarrow 0$ seems not easy proved from it.

My first question is how to prove $E|\xi_n-\xi|\rightarrow 0$.

And I don't know why should emphasize the condition "$\eta_n\le 0\le \zeta_n$", so my second question is: Is there any example that $E|\xi_n-\xi|\rightarrow 0$ is wrong if the condition is not fullfilled.

Thanks a lot, I'm looking forward to your answers!

Is there any example that $E|\xi_n-\xi|\rightarrow 0$ is wrong if the condition is not fullfilled.

Yes. Take your favorite example of centered random variables $\xi_n$ such that $\xi_n\to 0$ in probability but $\mathrm E|\xi_n|\not\to 0$ (e.g. $\xi_n = \pm n$ with probabilities $1/n$ and $0$ with probability $1-1/n$) and set $\eta_n = \zeta_n = \xi_n$.

Speaking about your first question, I would argue as follows: a sequence converges in probability iff any of its subsequences contains a subsubsequence converging almost surely, therefore, we can assume wlog that all convergences in question are almost sure. Then you just apply the Fatou lemma and conclude.