I am interested in how to prove the c.d.f. limit property that $lim_{x\rightarrow {-\infty}} F_x(x) =0$ for the generic case where we cannot assume that $x_n \rightarrow -\infty$ monotonically. I understand how to explain it intuitively but I am not sure that it qualifies as a proof.
I believe the reason that the limit is zero is that it is the equivalent of taking the probability of the set of events that is less than negative infinity. This probability must be zero since the set of such events will be empty.
$lim_{x\rightarrow {-\infty}}F_x(x) = P_x(A)$ where $A=\{x \leq -\infty\}= \emptyset$
$\Rightarrow P_x(A)=0$
Would this satisfy as a proper proof, or am I missing pieces? Thanks.
You can prove for some monotone decreasing sequence, say $-1, -2, -3, \ldots$, right? The limit $\lim\limits{n}_F(-n)= P(\cap{X<-n})= P(\emptyset)=0$.
The fact that $x_n\rightarrow -\infty$ means that for arbitrari(ly small) $r\in \mathbb{R}$ we have $x_n<r$ for $n$ large enough, say $n\geq n_r$. This means that if $n\geq n_r$, then $F(x_n)\leq F(r)$.
Put $r=-1, -2, -3, \ldots$, and you see that the limit $\lim\limits_n F(x_n)$ is also $0$.