My prof give us this exercise:
With these definitions:
$$E' := \liminf_{k\rightarrow \infty}\; E_k = \bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n$$
$$E'' := \limsup_{k\rightarrow \infty}\;E_k = \bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} E_n$$
Let $E_n = \left\{x \in [0,2\pi] : \dfrac{\sin(nx)}n > 0\right\}$ with $n\in \Bbb N$.
Calculate: $E', E''$
but in class he never did exemples on how to proceed in calculate limsup, liminf of subset sequences, he only gave us the definitions.
Could anyone explain me how to do this? (I posted this exemple to give you something concrete for a better explaining).
Thanks for your answers.