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Let V be a real vector-lattice and $u\geq 0, v\geq 0$.

I need to show that $[0, u+v] \subseteq [0,u]+[0,v]$.

I would appreciate a hint on how to start.

For $z\in [0,u+v]$ with $z\geq u$ or $z\geq v$ it is easy. My problem is that we can't assume $z<u$ and $z<v$ if the assumption above does not hold.

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Hint: Consider the element $\min(z,u) \in V$.

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  • $\begingroup$ Oh, is it min(z,u)+max(0,z-u)? I had something similar which didnt work.This one seems fine though or did i make a mistake? $\endgroup$
    – Lukas Betz
    Jun 10 '16 at 15:33
  • $\begingroup$ Yes, this belongs to $[0,u] + [0,v]$. $\endgroup$
    – gerw
    Jun 10 '16 at 17:17

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