Calculate $\lim \inf$ and $\lim \sup$ of $A_n$

Question

$A_n = (-1+\frac1n,2-\frac1n), $ for odd n,

$A_n = [0,n]$ for even n.

Calculate lim inf $A_n$ and lim sup $A_n$

I am unsure of how to go about calculating these limits. I know that $\liminf A_n = ⋃_m⋂_{(n≥m)}A_n$ but I am not sure how to apply this. Any help would be greatly appreciated!

Answer 1

The set $\liminf_nA_n$ is the set of those real numbers which belong to $A_n$, for every large enough $n$. So, $\liminf_nA_n=[0,2)$. In fact:

• if $x<0$, then $x$ belongs to no $A_n$ for even $n$;
• if $x\geqslant 2$, then $x$ belongs to no $A_n$ with odd $n$;
• if $x\in[0,2)$, then $x\in A_n$, for each $n>1$.

On the other hand, the set $\limsup_nA_n$ is the set of those real numbers which belong to infinitely $A_n$'s. Therefore, it is equal to $(-1,\infty)$.