Let $(f_n)_{n \in \mathbb{N}}$ a sequence of function from a nonempty-set $X$ to $\bar{\mathbb{R}}=[-\infty,\infty]$, $g\colon X \to \bar{\mathbb{R}}$ defined by $g(x)=\sup(n \in \mathbb{N}) f_n (x)$. Then $g^{-1}\bigl( (a,+\infty] \bigr)=\bigcup_{n \in \mathbb{N}} f^{-1}\bigl((a,+\infty]\bigr)$. Why?
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closed as too localized by Simon Thomas, Andres Caicedo, Zev Chonoles, Bill Johnson, Igor Rivin Oct 28 2011 at 17:18 |
