
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(Xnε>=0)>0 as n>inf. But P(Xnε>=0)=P(Xn>=ε)=P(Xn>=0)=P(Xnε>=ε) so this is true iff Xn>0 (in probability)
Is this logic correct?

Hello,
You misused the definition of convergence in probability.
$\displaystyle X_n\to X$ iff $\displaystyle P(X_nX>\epsilon)\to 0$
So here it gives $\displaystyle P(X_n>\epsilon)\to 0$
If you try to write it, this is exactly the definition of $\displaystyle X_n$ converging in probability to 0.