Folland defines $$\limsup E_n=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}E_n,\liminf E_n=\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}E_n.$$ And states that $\limsup E_n=\{x:x\in E_n$ for infinitely many $n$}, $\liminf E_n= \{x:x\in E_n$ for all but finitely many $n$}
Is not "all but finitely many"="infinite"?
$\{E_n\}$ is infinite family.
The set of even numbers is infinite, but it is not "all but finitely many" of the integers. So: let $E_n = \{1,2\}$ for all even $n$ and $E_n = \{2,3\}$ for all odd $n$, and try computing liminf and limsup.