Vector lattice: $[0,v]+[0,u] = [0,u+v]$

Question

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.

Answer 1

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