Could someone please explain why the following hold for a set of a functions $f_n(x)$:
$$\{x : \sup f_n > c \} = \bigcup_{n=1}^\infty \{ x: f_n(x) > c \}$$ $$\{x: \inf f_n < c \} = \bigcup_{n=1}^\infty \{ x: f_n(x) < c \}$$
thank you
2026-04-04 12:08:56.1775304536
Writing $\sup$ and $\inf$ in unions
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If $\sup f_n (x) > c$, then there exists $n \in \mathbb N$ such that $f_n(x) > c$ (if not, then $f_n(x) \leq c, \forall n$, which implies $\sup f_n(x) \leq c$, contradiction). So we have: $$\sup f_n(x) > c \Leftrightarrow \exists n \in \mathbb N: f_n(x) > c $$ Thus, $$x \in \{\sup f_n > c \} \Leftrightarrow \exists n \in \mathbb N: x \in \{f_n > c \} \Leftrightarrow x \in \cup_n\{f_n > c\}.$$ Similar for $\{\inf f_n < c \}$.