2
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I am trying to proof the same as the question bellow, but for bounded functions over a field $\mathbb{K}$. p is defined the same way as the question but we are taking the limit of a function when it's variable goes to $\infty$ enter image description here

Generalized limit in $l_\infty$ (Using: Hahn Banach Extension Theorem)

I have no problem with the further proof, after definition of $p$, but I'm not getting how to prove the following:

  1. p is well defined in $\mathcal{L} _{\infty}$
  2. p is subadditive
  3. if $a \leq \phi(x) \leq b, \forall x \in \mathbb{R}$ then $a \leq p(\phi) \leq b $
  4. $p(\phi(s) - \phi(s+a)) = 0, \forall \phi \in \mathcal{L}_{\infty}$
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    $\begingroup$ If you can't even prove (1) then you are in serious trouble. Recall $x\in\ell_\infty$ means there is an $M\in\mathbb{R}$ such that $\lvert x_m\rvert<M$ for every $m$. Now use triangle inequality then take limsup and inf... $\endgroup$ – user10354138 May 14 '19 at 18:52
  • $\begingroup$ @user10354138 I shouldn't have put (1), thanks for the answer. It is quite easy using the fact that $\phi \in \mathbb{L}_{\infty}$. I made (2) until a certain extend, where I would have the wrong side of the inequality with the inf. And I'm having a lot of trouble with (4), even though it seems intuitive. $\endgroup$ – Rudá Lima da Floresta May 14 '19 at 19:00

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