Let $O$ be some set and $A_n, B_n \subset O$ sequences of subsets. If $A_n \subset B_n$ for all $n$ then if $\lim\limits_{n \to \infty} A_n, \ \ \lim\limits_{n \to \infty} B_n$ exist, is $\lim A_n \subset \lim B_n$?
Do we need more information about what $\lim A_n$ is?
As $\lim A_n$ and $\lim B_n$ exist, we have:
$$\lim A_n = \lim \sup A_n= \bigcap_{n=1}^{\infty} \bigcup_{k\ge n} A_k$$
$$\lim B_n = \lim \sup B_n = \bigcap_{n=1}^{\infty} \bigcup_{k\ge n}B_k$$
As for all $n$, $A_n \subset B_n$, we have for all $n$,
$$\bigcup_{k\ge n} A_k \subset \bigcup_{k\ge n} B_k$$
Therefore the result.
If $A_n\subseteq B_n$ for each $n$ then you can conclude:$$\limsup A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k\subseteq\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}B_k=\limsup B_n$$
If $\lim_{n\to\infty}A_n$ and $\lim_{n\to\infty}B_n$ both exist then they coincide with $\limsup A_n$ resp. $\limsup B_n$
