I am trying to prove the following:
Let $(A_n)_{n \geq 1}$ be a sequence of measurable sets, then
(i) $|\lim \inf_{n \to \infty} A_n| \leq \lim \inf_{n \to \infty} |A_n|$
(ii) If there is $n \in \mathbb N$, $|\bigcup_{k=n}^{\infty} A_k|< \infty$, then $|\lim_{n \to \infty} \sup A_n| \geq \lim_{n \to \infty} \sup |A_n|$.
I got stuck trying to show both inequalities.
In (i), on one hand we have
$$|\lim \inf_{n \to \infty} A_n|=|\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k|$$$$\leq \sum_{n=1}^{\infty} |\bigcap_{k=n}^{\infty}A_k|$$$$\leq\sum_{n=1}^{\infty}\sup_{k \leq n} |A_k|$$
In (ii), we have $$\lim_{n \to \infty} \sup |A_n|=\inf_{n \geq 0} \sup_{k \geq n}|A_k|$$$$\leq \sup_{k \geq n}|A_k| \space \forall n$$
I would appreciate some help where I got stuck, thanks in advance.