I'm struggling to understand the beginning of the solution to the following exercise:
Let $(X_n)_{n\geq 1}$ and $X$ be random variables. Prove that $X_n \to X$ almost surely if and only if for every $\epsilon>0$ $$P(\limsup\limits_{n\to \infty}\{|X_n-X|\geq \epsilon\})=0$$
Solution:
Notice that $\omega \in\limsup\limits_{n\to \infty}\{|X_n-X|\geq \epsilon\}$ iff there exist a subsequence $(n_k)_{k\geq 1}$ such that $|X_{n_k}-X|\geq \epsilon$, so we have $$\{\limsup\limits_{n\to \infty}|X_n-X|> \epsilon\}\subset \limsup\limits_{n\to \infty}\{|X_n-X|\geq \epsilon\} \subset \{\limsup\limits_{n\to \infty}|X_n-X|\geq \epsilon\}\quad (*)$$
and so on..
I don't understand what this means.
- First what subsequence are we referring to? A subsequence that does converge? Why do we need this exactly?
- What's the difference between those 3 sets in line $(*)$. Why does it matter if the $\limsup$ is within the brackets and why does it matter whether $>$ or $\geq$? I really don't understand what the difference are.
I'm happy if someone could explain it to me.