Is the following true:
$\sup(f+g)\ge \sup(f)+\inf (g)$?
If so prove it.
I think its true, but I don't know how to prove it
2025-01-12 23:58:20.1736726300
Proof of supremum and infemum of bounded functions
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Take an arbitrary $x$ in the domain of the functions, so that: $$f(x) + \inf g \leq f(x) + g(x) = (f+g)(x)\leq \sup (f+g), \quad \forall x.$$This directly yields $\sup f + \inf g \leq \sup(f+g)$. You can think that: $$f(x) \leq \sup(f+g) - \inf g, \quad \forall x,$$so you can take $\sup$ in the left side and then rearrange.