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### Proving $\sup(A+B)=\sup(A)+\sup(B)$ [duplicate]

Possible Duplicate: How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B= \lbrace a+b\mid a\in A, b\in B\rbrace$ here's a homework question I'm currently working on: Let $A,B \subset \mathbb{R}$ ...
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### Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$ [duplicate]

Let $A,B$ be subsets of $\mathbb{R}$. Prove that $\sup{(A+B)} = \sup{A}+\sup{B}$. I think in order to solve this we are going to have to use the mathematical definition of supremum. Maybe we can ...
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### Why is $L(P,f)+L(P,g) \le L(P,f+g) \le U(P,f+g) \le U(P,f)+U(P,g)$
Let $R \subset \Bbb{R}^n$ and $f,g$ be two integrable functions from $R\to \Bbb{R}$ , in a proof of a property of integrals Rudin writes the following: L(P,f)+L(P,g) \le L(P,f+g) \le U(P,f+g) \le U(...
In this question $\mathcal{H}$ stands for a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let \$\mathcal{B}...