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I need help with these:

1) Show the following inequality for the supremum of functions $f:\mathbb{R}\to \mathbb{R}$ and $g: \mathbb{R}\to\mathbb{R}$ $$ \sup(f+g)(x)\leq \sup f(x)+\sup g(x) $$

2) What can you say about a set $M$ of real numbers if you know that $\sup M = \inf M$?

Could you please, before you just giving the proof write down the definitions you are using. I do not want the direct proof, only how I should think and what definitions I should use, so I can try to proof it by myself before I see your guys proof :)

Thanks a lot!

Stefan Hansen
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Fred
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  • See here. – J.R. Jan 29 '14 at 12:38
  • Sorry. What can you say about a set M if you know that sup M = inf M?

    Show the following inequality for the supremum of functions f: R-> R and g: R->R: sup(f+g)(x)≤supf(x)+supg(x)

     – Fred Jan 29 '14 at 12:39
  • If you have details to add to your qeustion, please add them by editing your question and not commenting. – J.R. Jan 29 '14 at 12:42
  • Did you by $M$ mean a set of real numbers or the set of all pink Afghan elephants in blue pyjamas somehow equipped with a total ordering? – J.R. Jan 29 '14 at 12:49
  • What are your definitions of $\sup$ and $\inf$? – user37238 Jan 29 '14 at 12:52
  • i did just copy paste the question. But i will change it to real numbers – Fred Jan 29 '14 at 12:53

1 Answers1

5

(1)

$\forall x\ f(x)\le\sup f$ ($\sup$ is upper bound)

$\forall x\ g(x)\le\sup g$ (idem)

$\forall x\ f(x)+g(x) ≤ \sup f+\sup g$ (add inequalities)

$\sup f+g ≤ \sup f+\sup g$ (because $\sup$ is the least upper bound)

(2) Your "amount" is "set"?

Martín-Blas Pérez Pinilla
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  • thanks for the quick reply. yes, I meant set. – Fred Jan 29 '14 at 12:45
  • Find an example to show that a strict inequality that may occur.

    I took 4sin(x) as f and 3cosx as g. Is this correct?

    f(x)+g(x)=5sin(x+y) has sup(f+g)(x)=5 which is strict inequality with supf+supg = 7.

     – Fred Jan 29 '14 at 16:16