I'm working on the following problem for a probability class.

Let $\displaystyle X_1, X_2,...$ be independent random variables with a common continuous distribution function. Let $\displaystyle B$ be the $\displaystyle \omega$-set where $\displaystyle X_m(\omega)=X_n(\omega)$ for some pair $\displaystyle m,n$ of distinct integers, and show that $\displaystyle P(B)=0$.

Part of my problem is that this section we're studying on random variables introduces so many new terms that I haven't been able to see how they work together very well. I understand that $\displaystyle P(B)=0$ means that $\displaystyle X_m(\omega)\ne X_n(\omega)$ for any $\displaystyle m\ne n$ except on a set of probability (measure) 0, but am struggling mostly with the first sentence. I'm not asking for a full fledged solution to the problem, but I would be grateful for any assistance on understanding the interplay between the other terms involved and advice on how to start.