Set Inclusion Properties

Question

Consider a sequence of random variables $(X_n)$ converge almost surely to $X$. Define set $N:=\{\omega: X_n \to X \}^C$. Then it is claimed that we would have the following set inclusion properties.

$$ N^C \cap \{\omega: X \leq x-h\} \subset \liminf \{\omega: X_n\leq x\}\cap N^C \subset \limsup \{\omega: X_n\leq x\}\cap N^C \subset \{\omega: X\leq x\}\cap N^C. $$

I can understand the second inclusion, but I do not get the first and the last one. Could anyone explain to me, please? Thank you!

Answer 1

Assume $h>0$.

For the first, we have

$$ \begin{align} N^C\cap\{X\leqslant x-h\} &= N^C\cap\{\lim\sup X_n\leqslant x-h\}\ \subseteq\ N^C\cap\{\lim\sup X_n\leqslant x\} \\ &= N^C\cap\bigcup_{n\geqslant 1}\bigcap_{k\geqslant n}\{X_k\leqslant x\}\ =\ N^C\cap\lim\inf\{X_n\leqslant x\} \end{align} $$

So,

$$ N^C\cap\{X\leqslant x-h\}\ \subseteq\ N^C\cap\lim\inf\{X_n\leqslant x\} $$

And, for the last,

$$ \begin{align} N^C\cap\lim\sup\{X_n\leqslant x\} &= N^C\cap\bigcap_{n\geqslant 1}\bigcup_{k\geqslant n}\{X_k\leqslant x\} \ \subseteq\ N^C\cap\{\lim\inf X_n\leqslant x\} \\&= N^C\cap\{X\leqslant x\} \end{align} $$

So,

$$ \begin{align} N^C\cap\lim\sup\{X_n\leqslant x\}\ \subseteq\ N^C\cap\{X\leqslant x\} \end{align} $$