Example of sequences of subsets where $\liminf (A_n) = 0$ and $\limsup (A_n) = X$

Question

So $X$ is a (fixed, unknown) set and $A\subset X$ be a proper subset.

I need to come up with examples of sequences $(A_n)_{n \in \Bbb N}$ of subsets of $X$ such that,

$\liminf (A_n) = 0$ and $\limsup (A_n) = X$

So far, I have $$A_{2n}= B$$ and $$A_{2n+1}= X\setminus B$$ where $B\neq \emptyset \neq X$

as one example. Can anyone tell me is this correct and give me any other examples?

Answer 1

I assume what you mean by $B \neq \emptyset \neq X$ is that $B,X,\emptyset$ are pairwise distinct. In that case yes, it is correct, but you can drop the condition of $B \neq \emptyset$. That is, you can simply let $A_{2n} = X$ and $A_{2n+1} = \emptyset$, and your conditions would be satisfied.

Answer 2

This is correct (the proper set involved is denoted by $A$ instead of $B$).

For other examples, let $I$ be a subset of $\mathbb N$ such that $I$ and $\mathbb N\setminus I$ are both infinite. Let $B_n=A$ if $n\in I$ and $B_n=X\setminus A$ if $n\notin I$.