$\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}=\{\lim _ {n\rightarrow \infty }\sup f_n \leq t \}$ Proof and Intuition

Question

For a start how can I read $\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}$ ? I only seen $\lim _ {n\rightarrow \infty }\inf $ for sets and just don't understand the meaning of $\lim \inf$ for an inequality such as $\{f_n \leq t\}$

How could the following be proved? thank you

$$\lim _ {n\rightarrow \infty }\inf \{f_n \leq t \}=\{\lim _ {n\rightarrow \infty }\sup f_n \leq t \}$$

Answer 1

Some people use "$\{ f_n \le t \}$" as short-hand for "$\{ x : f_n(x) \le t \}$".

Thus the statement you're asking about probably means:

$\lim_{n\to\infty} \bigcap_{m>n} \{ x : f_n(x) \le t \} = \{ x : \limsup_{n\to\infty} f_n(x) \le t \}$.