I am reading measure theoritic probability with self study mode from online materials. In limsup and liminf of sequence of sets I am confused about how to tackle this question.
Given $\Omega$ is a nonempty set and $A,B \subseteq\Omega$ with $A \neq B$.
Define, $A_{n} = \begin{cases} A & \text{if n is odd}\\ B &\text{if n is even} \end{cases}$.
How to find $\limsup A_{n}$ and $\liminf A_{n}$.
My approach is to use the formal definition which uses the union and intersection of sets.Very much depressed from that. How should I solve this. Any explanatory proof is highly appreciated. Thanks in advance.
No depression is required. $$ \liminf A_n = \bigcup _{k=1}^\infty \bigcap _{n\geqslant k}A_n \overset{?}= A\cap B $$ For $\subseteq$ take any $k\in\mathbb N$ and note that $\omega\in \bigcap _{n\geqslant k}A_n$ implies $\omega \in A\cap B$. Conversely, $\omega\in A$ means that $\omega \in A_m$ for all odd $m\in\mathbb N$. So, assuming $\omega\in A\cap B$, there must exist $k\in\mathbb N$ such that $$ \forall n\left ( n\geqslant k \Rightarrow \omega \in A_n \right ). $$ The argument for the upper limit works out similarly.