If independent Xn converge in probability they almost surely converge to a constant

Hi guys,

I hope you might be able to help me with this question:

Suppose $\displaystyle X_n$ are independent, and $\displaystyle X_n \rightarrow X$ in probability. Show that $\displaystyle X$ is almost surely a constant.

The hint I got was: Suppose that $\displaystyle X$ is not a constant almost surely. Can there be $\displaystyle a$ and $\displaystyle b$ with $\displaystyle a<b$, such that P(X<a)>0 and P(X>b)>0 ? Now use independence of $\displaystyle X_n$ and X_n+1.

I really have no idea where to go with this so any help would be much appreciated! PS. Sorry about the bits not written out in Latex - it kept saying Latex compile error whenever I submitted them.

Many thanks,

YChana