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#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

Let $S$ be a non-empty bounded set of real numbers, and $\overline{m}=\sup S$. Prove that $\inf \{\overline{m}-x: x\in S\}=0$.

[Use only the definitions of supremum and infimum, and not identities like $\inf(A+B)=\inf A+\inf B$ and $\inf(-S)=-\sup(S)$.]

[Use only the definitions of supremum and infimum, and not identities like $\inf(A+B)=\inf A+\inf B$ and $\inf(-S)=-\sup(S)$.]

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