# Math Help - probability convergence

1. ## probability convergence

Show that Xn->0 (in probability) iff |Xn|->0 (in probability).

What i have so far
|Xn|->0 (in probability) iff for any ε>0 P(||Xn|-ε|>=0)->0 as n->inf. But P(||Xn|-ε|>=0)=P(|Xn|>=ε)=P(|Xn|>=0)=P(|Xn-ε|>=ε) so this is true iff Xn->0 (in probability)

Is this logic correct?

2. Hello,

You misused the definition of convergence in probability.

$|X_n|\to X$ iff $P(||X_n|-X|>\epsilon)\to 0$
So here it gives $P(|X_n|>\epsilon)\to 0$

If you try to write it, this is exactly the definition of $X_n$ converging in probability to 0.