Hello,

By Borel-Cantelli's lemma (part I), we have $\displaystyle \displaystyle P(\limsup_n \{|X_n-X|<\epsilon\})=0$. By taking the complement, we get that $\displaystyle \displaystyle \forall \epsilon>0,P(\liminf_n \{|X_n-X|<\epsilon\})=1$

This means that $\displaystyle \displaystyle\forall \epsilon>0,P(\exists N\in\mathbb N,\forall n\geq N,|X_n-X|<\epsilon)=1$.

Then you have to get the epsilon into the probability, and I think you have to assume first that epsilon is rational (so that there is a countable number of epsilon's).

I don't really remember how to do and I have to go out right now, I'll try to think about it, unless you do before me