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Why $|f(x)|\leq C,\forall x\in \mathbb{R}$ if and only if $\sup_{x\in\mathbb{R}}|f(x)|\leq C$?

It's clear that $|f(x)|\leq \sup_x|f(x)|\leq C$.

But why we are allowed to take supremum of both sides? By definition $\sup_{x\in\mathbb{R}}|f(x)| = \sup\{f(x):\; x\in\mathbb{R}\}$, but I fail to see this.

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    $\begingroup$ Because $\sup_x f(x) \ge f(y)$ for all $y$. $\endgroup$
    – user207868
    Commented Nov 13, 2015 at 15:20

1 Answer 1

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The statement that $|f(x)| \le C$ for all $x \in \mathbb R$ means that $C$ is an upper bound of the set $\{|f(x)| : x \in \mathbb R\}$. Thus it is either equal to, or greater than, the least upper bound: $$\sup \{|f(x)| : x \in \mathbb R\} \le C.$$

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