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Given two real-valued functions $f$ and $g$, is it true that $\sup(f-g) \geq \sup(f) - \sup(g)$

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Using the argument in this answer, we have $$ \sup((f-g)+g)\le\sup(f-g)+\sup(g) $$ which becomes $$ \sup(f-g)\ge\sup(f)-\sup(g) $$

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