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Suppose $f : A \to \mathbb{R}$ and $g: A \to \mathbb{R}$ are real valued functions. Define $(f+g)[A] = \{f(x) + g(x): x \in A\}$ and $f[A] + g[A] = \{f(x) + g(y): x, y \in A\}$.

What is the relationship between $\sup(f[A] + g[A])$ and $\sup((f+g)[A])$?

Repeat exercise for $\inf(f[A] + g[A])$. Essentially one side is $f(x) + g(x)$ and the other is $f(x) + g(y)$. I am thinking they are equal, but I'm not sure. I need a proof too.

user66733
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MDW
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2 Answers2

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use that $ \{ f(x) + g(x): x \in A\} \subset \{f(x) + g(y): x, y \in A\}$

helmonio
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So we know that, using your definitions, $$(f+g)[A] \subseteq f[A]+g[A]$$ because every element of the right side is an element of the left by setting $y=x$. Therefore, any upper bound for the left side is also an upper bound for the right, and we have $$\sup((f+g)[A]) \leq \sup(f[A]+g[A])$$

The dual calculation applies for the infimum (can you see why?) and we get that $$\inf((f+g)[A]) \geq \inf(f[A]+g[A])$$

guest196883
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