Our tutor claimed that $\{x\in M\mid\liminf\limits_{n\to\infty}f_n(x)\geq 1\}=\bigcup\limits_{n=1}^{\infty}\bigcap\limits_{k=n}^{\infty}\{x\in M\mid f_n(x)\geq 1\}$, where $f_n:M\to\mathbb{R}$.
But I think that it is a typo and instead it should be
$$\bigcup\limits_{n=1}^{\infty}\bigcap\limits_{k=n}^{\infty}\{x\in M\mid f_n(x)\geq 1\}\subseteq\{x\in M\mid\liminf\limits_{n\to\infty}f_n(x)\geq 1\}.$$
$f_n(x)=1-\frac{1}{n}$ should serves as an easy counter-example. However, my tutor didn't pay much attention to my concerns. Did I miss something?