Inequalities related to infimum and supremum

Question

Let $f,g: A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$. Let $P$ be a partition of $A$ and $S \in P$ a sub-rectangle. Show that:

$m_S(f+g) \geq m_S(f)+m_S(g)$ and $M_S(f+g) \leq M_S(f)+ M_S(g)$

Answer 1

Ok at first we will prove that $$\inf_{S}(f+g)\geq \inf_S f +\inf_S g$$

For this we use that $$g\geq \inf_S g$$ because of the definition of infimum. Hence $$\inf_S ( f+ g) \geq \inf_S( f+\inf_S g)$$ On the other hand $\inf_S g$ is only a constant, hence $$\inf_S (f+\inf_S g)=\inf_S f + \inf_S g$$

Now we prove that $$\sup (f+g)\leq \sup f+\sup g.$$ The proof is nearly the same we use that $$g\leq \sup g$$ because of the definition of supremum. Hence $$\sup(f+g)\leq \sup(f+\sup g)$$ As $\sup g$ is only a constant we get again $$\sup(f+\sup g) =\sup f +\sup g $$ Hence $$\sup (f+g) \leq \sup f + \sup g$$