\begin{proof} The middle inequality holds by definition so we prove only the first inequality since the last holds using the same logic.

First recall that $$\liminf_{ n \to \infty }A_{n}=\bigcup_{n\ge 1} \bigcap_{k\ge n} A_{k}$$Then note that $\bigcap_{k\ge n}A_{k}$ is a sequence of increasing events in $n$ to $\liminf_{ n \to \infty }A_{n}$, hence by Continuity of Probability we have $$\mathbb{P}\left(\liminf_{ n \to \infty } A_{n}\right)=\mathbb{P}\left( \bigcup_{n}\bigcap_{k\ge n}A_{k} \right)=\lim_{ n \to \infty } \mathbb{P}\left( \bigcap_{k\ge n} A_{k} \right)=\liminf_{ n \to \infty } \mathbb{P}\left( \bigcap_{k\ge n}A_{k} \right)$$and since $A_{n}\supseteq\bigcap_{k\ge n}A_{k}$, by Probability Measure we have $$\liminf_{ n \to \infty } \mathbb{P}\left( \bigcap_{k\ge n} A_{k}\right)\le\liminf_{ n \to \infty } \mathbb{P}(A_{n}).$$

\end{proof} ## Remark The Limits of Events is useful for characterizing “rare events” in a stochastic process or for understanding the long-term behavior of random systems.

In simpler terms, it can answer questions like, “Given a random process, what can we say will eventually happen with certainty?” This is crucial for understanding phenomena where we’re not just interested in immediate or short-term randomness but also in the behavior of the system over an extended period.