# Math Help - Expectation in probability convergence

1. ## Expectation in probability convergence

Prove that $X_n$ converges to 0 in probability if and only if $E ( \frac { |X_n | } {1 + |X_n|}) \rightarrow 0$

So I know that $P(|X_n-0| \geq \epsilon ) \rightarrow 0$, but I don't really know how I should rewrite the expectation value part to make this work. Thank you!

2. ## Re: Expectation in probability convergence

You have to use the fact that the map $x\mapsto\frac x{x+1}$ is increasing. If we assume that $E\left[\frac{|X_n|}{1+|X_n|}\right]\to 0$, for a fixed $\varepsilon$, $\frac{\varepsilon}{1+\varepsilon} P(|X_n|\geq \varepsilon)$ can be bounded above by $E\left[\frac{|X_n|}{1+|X_n|}\right]$.