MHB Is the sequence $(X_n)$ of $L^1$ random variables uniformly integrable?

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    2017
Euge
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Here is this week's POTW:

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Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that

1. $\Bbb E[f(|X_n|)]$ is uniformly bounded

2. $\dfrac{f(x)}{x}\to \infty$ as $x\to \infty$

Show that $(X_n)$ is uniformly integrable.
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Due to the difficulty some may have had with this problem, I'm extending the deadline one more week.
 
No one answered this week's problem. You can read my solution below.
Set $M = \sup_{m\in \Bbb N} E[f(\lvert X_n\rvert)]$. Given $\epsilon > 0$, choose $\delta > 0$ such that for all $x$, $x \ge \delta$ implies $f(x) > \frac{x}{\epsilon}$. For all $n\in \Bbb N$,

$$E[\lvert X_n\rvert I_{\lvert X_n\rvert \ge \delta}] \le E[\epsilon f(\lvert X_n\rvert)I_{\lvert X_n\rvert \ge \delta}] \le \epsilon M$$

Since $\epsilon$ was arbitrary, $(X_n)$ is uniformly integrable.
 
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