Limsup of a sequence smaller than $x$

Question

Let $X_n$ be a sequence of random variables. Show that:

$$ \{\limsup X_n < x\} = \bigcup_{k=1}^{\infty} \bigcup_{n=1}^{\infty} \bigcap_{ m =n }^{\infty} \{ X_m < (x - (1/x)) \} $$

First question :the def of limsup says that it is the intersection of the union. I don't get why above the final statement is written as the union of the union of the intersection.
Second question: I don't know where the first Union going from k to inf comes from.
Third question: What's the " 1/x" and how is it interpretable?

Answer 1

The correct statement should be $$ \{\limsup X_n < x\} = \bigcup_{k=1}^{\infty} \bigcup_{n=1}^{\infty} \bigcap_{ m =n }^{\infty} \{ X_m < (x - \frac{1}{\color{red}{k}}) \}\,. $$

• An $\omega\in\Omega$ is in the RHS if and only if there exists a $k\ge 1$ and an $n\ge 1$ such that for all $m\ge n$ $$ X_m(\omega)<x - \frac{1}{\color{red}{k}}\,. $$
• This is equivalent to saying that $$ \limsup_nX_n(\omega)<x\,. $$