Proposition 8 Chapter 3 Real Analysis Royden

Question

Underlined equality is not correct I believe:

r.h.s. means the set of all $x$;s (in $E$) s.t. $f_1(x)>c$ 'in addition' to the all $x$;s (in $E$) s.t. $f_2(x)>c$, ... (all up to $k=n$); and in general case at least one of $f$;s is not equal to the $\max {\{f_1, \dots, f_n}\}$ so implies $\subset$. So r.h.s. contains l.h.s. but not the reverse. Am I wrong?

Answer 1

You are wrong; the equality is valid.

If $x$ is in the set on the RHS, then there is some $k$ such that $f_k(x)>c$. Since $\max\left\{f_i\right\}(x)\geq f_k(x)$, it follows that $\max\left\{f_i\right\}(x)>c$, so $x$ is an element of the LHS.

Answer 2

$x\in \bigcup_{k=1}^n \{f_k > c\}$ $\iff$ $x$ is in at least one $\{f_{k^*} >c\}$ $\iff$ $x\in \{\max \{f_1,...f_n\} >c\}$