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I know that , for any bounded sets $A,B\subset \mathbb{R}$, $\sup A + \sup B = \sup (A+B)$. But what if $A,B$ are not arbitrary, but functions? Let $f_i:[a,b]\rightarrow \mathbb{R}$ be bounded functions and $\alpha:[a,b]\rightarrow \mathbb{R}$ be a monotonically increasing function and $P$ be any partition of $[a,b]$. How do i prove that $L(P,f_1,\alpha) + L(P,f_2,\alpha) ≦ L(P,f_1 + f_2, \alpha)$? |
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I think this question is asking how to show something like $$ \sup_{x\in A}(f(x)+g(x))\le\sup_{x\in A}f(x)+\sup_{x\in A}g(x)\tag{1} $$ This is an instance of the fact that the supremum over a set is no smaller than the supremum over a subset. The left hand side of $(1)$ is $$ \sup_{\substack{x,y\in A\\x=y}}(f(x)+f(y))\tag{2} $$ whereas the right hand side of $(1)$ is $$ \sup_{x,y\in A}(f(x)+f(y))\tag{3} $$ The set of $x$ and $y$ being considered in $(2)$ is a subset of the $x$ and $y$ being considered in $(3)$, so $(1)$ follows. |
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(real-analysis)tag. – kahen Nov 25 '12 at 16:51