Let $X_n,n\geq 1$ be a sequence of real random variables defined on a probability space. Show that the following are equivalent:
(i)$X_n$ converges in probability to a real random variable $X$.
(ii) For every $\epsilon>0,P(|X_m-X_n|>\epsilon)\rightarrow 0$, as $n,m\rightarrow\infty$
(iii) For every $\epsilon>0,sup_{m>n}P(|X_m-X_n|>\epsilon)\rightarrow 0$, as $n\rightarrow\infty$
I have been able to show (i) implies (ii) and (ii) implies (iii) and I am struggling with (iii) implies (i). (iii) implies (ii) will also do.The main obstruction is that from the sequence I am not being able to get hold of any convergent sequence $X$. I thought of using completeness of $L_1$, then again only convergence in probability does not imply convergence in $L_1$. Please help.