$\liminf A_n$ and $\limsup B_n$

Question

Show that for a sequences of sets $A_n$ and $B_n$

$\liminf A_n\cap\limsup B_n \subset\limsup(A_n\cap B_n)$

Can you give some hint please.How can ı show this question.

Thank you

Answer 1

Let $x\in\liminf A_{n}\cap\limsup B_{n}$.

$x\in\liminf A_{n}$ tells us that some $n_{0}\in\mathbb{N}$ exists such that $n\geq n_{0}\implies x\in A_{n}$

$x\in\limsup B_{n}$ tells us that a sequence $\left(n_{k}\right)$ exists in $\mathbb{N}$ with $n_{1}<n_{2}<\cdots$ and $x\in B_{n_{k}}$ for each $k$.

We can choose $n_{1}\geq n_{0}$ and then also $x\in A_{n_{k}}\cap B_{n_{k}}$ for each $k$, telling us that...