[proofplan] Write the limsup and liminf as countable combinations of the sets $A_n$ and use closure of the sigma-algebra under countable unions and intersections. [/proofplan] [step:Write limsup] The limsup is \begin{align*} \limsup_{n\to\infty}A_n=\bigcap_{m=1}^{\infty}\bigcup_{n=m}^{\infty}A_n. \end{align*} For each $m$, the union $\bigcup_{n=m}^{\infty}A_n$ belongs to $\mathcal F$. Taking the countable intersection over $m$ keeps the set in $\mathcal F$. [/step] [step:Write liminf] The liminf is \begin{align*} \liminf_{n\to\infty}A_n=\bigcup_{m=1}^{\infty}\bigcap_{n=m}^{\infty}A_n. \end{align*} For each $m$, the intersection $\bigcap_{n=m}^{\infty}A_n$ belongs to $\mathcal F$. Taking the countable union over $m$ keeps the set in $\mathcal F$. [/step] [step:Conclude the proof] Both $\limsup_{n\to\infty}A_n$ and $\liminf_{n\to\infty}A_n$ are elements of $\mathcal F$. [/step]