Generalized limit in $\mathcal{L} _{\infty}$ (Using: Hahn Banach Extension Theorem)

Question

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 $\infty$

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. $p(\phi(s) - \phi(s+a)) = 0, \forall \phi \in \mathcal{L}_{\infty}$