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**Connected** Let $\displaystyle f,g:A\to\mathbb R$ be bounded above, then $\displaystyle f+g$ is bounded above and $\displaystyle \sup(f+g)\le \sup f+\sup g.$

Okay, I think there's no much to do here, we have $\displaystyle f\le \sup f$ and $\displaystyle g\le \sup g$ so $\displaystyle f+g$ is bounded above and besides $\displaystyle f+g\le \sup f+\sup g\implies \sup(f+g)\le \sup f+\sup g.$ Is it correct?

Now if $\displaystyle g$ is bounded below, then $\displaystyle \sup f+\inf g\le \sup(f+g).$

Well since $\displaystyle \inf g\le g\implies f+\inf g\le f+g$ so $\displaystyle \sup f+\inf g\le\sup(f+g),$ and we're done. Is it correct?