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#### Alexmahone

##### Active member

- Jan 26, 2012

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(a) Let $S$ be a bounded non-empty subset of $\mathbb{R}$, and $\overline{m}=\sup S$. Prove there is a sequence $\{a_n\}$ such that $a_n\in S$ for all $n$, and $a_n\to\overline{m}$. (You must show how to construct the sequence $a_n$.)

(b) Let $A$ and $B$ be bounded non-empty subsets of $\mathbb{R}$. Prove the equality $\sup (A+B)=\sup A+\sup B$. (Use part (a).)

(b) Let $A$ and $B$ be bounded non-empty subsets of $\mathbb{R}$. Prove the equality $\sup (A+B)=\sup A+\sup B$. (Use part (a).)

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