Hello,
You misused the definition of convergence in probability.
iff
So here it gives
If you try to write it, this is exactly the definition of converging in probability to 0.
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?