Hey,

$\displaystyle (A_{n})_{n \geq 1} $ a sequence of subsets of $\displaystyle \mathcal{A} $, a $\displaystyle \sigma $-field on $\displaystyle \Omega $.

With

$\displaystyle \lim_{n \to \infty} \inf A_{n} = \bigcup_{n=1}^{\infty} \bigcap_{m\geq n} A_{m} $

and

$\displaystyle \lim_{n \to \infty} \sup A_{n} = \bigcap_{n=1}^{\infty}\bigcup_{m \geq n} A_{m} $

show that $\displaystyle \lim_{n \to \infty} \inf A_{n} \subset \lim_{n \to \infty} \sup A_{n} $

Hints on how to start?