I'm confused about a conclusion in a proof:
Suppose we have a sequence $(X_n)$ of r.v. with values in $\mathbb{N}$, which onverges to a r.v. $X\in[0,\infty)$ a.s. Therefore for large $n$ we know $X_n=X$.
Further suppose for any $l\ge1$ and $N\ge0$ we have:
$P\left(\bigcap_{n\ge N}\{X_n=l\}\right) =0$
Now the question is, why can I deduce from this facts, that $P(X\ge1)=0$?
Thanks in advance for your help
hulik