Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Let $X_1, X_2, ...$ be a sequence of real-valued random variables. 
Prove $X_n \xrightarrow P  0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$
Attempt:
Suppose $X_n \xrightarrow P  0$ as $n \rightarrow \infty$. Then since $X_1, X_2, ...$  is uniformly integrable and E(|X|)<$\infty$, then $\lim_{n \to \infty} E(X_n) = E(X)$.
Since $X_n \xrightarrow P  0$ as $n \rightarrow \infty$, then $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )$ must be equal to 0.

Hint:
$X_n \xrightarrow P  0$ mean $\forall \epsilon >0, P(|X_n| > \epsilon) \to 0$.
$\frac{|X_n|}{|X_n|+1}  = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_n|}{|X_n|+1} 1_{|X_n| \leq \epsilon} \leq 1_{|X_n| > \epsilon} + \epsilon$
$\frac{|X_n|}{|X_n|+1}  = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_n|}{|X_n|+1} 1_{|X_n| \leq \epsilon} \geq 1_{|X_n| > \epsilon} \frac{\epsilon}{1+\epsilon}$
