Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Question

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.

Answer 1

Hint:

$X_n \xrightarrow P 0$ mean $\forall \epsilon >0, P(|X_n| > \epsilon) \to 0$.

$\frac{|X_n|}{|X_n|+1} = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_n|}{|X_n|+1} 1_{|X_n| \leq \epsilon} \leq 1_{|X_n| > \epsilon} + \epsilon$

$\frac{|X_n|}{|X_n|+1} = \frac{|X_n|}{|X_n|+1} 1_{|X_n| > \epsilon} + \frac{|X_n|}{|X_n|+1} 1_{|X_n| \leq \epsilon} \geq 1_{|X_n| > \epsilon} \frac{\epsilon}{1+\epsilon}$