Sum of functions, maximum property

Question

so i came along the following post Maximization of sum of two functions

and I would like to know how for any given functions $f(x),g(x)$ we can prove that:

$\max(f+g)<=\max f+ \max g$

I do have a small hint about triangle inequality but i am not sure. Any ideas?

Answer 1

Note that by definition

• $f \le \max(f)$
• $g \le \max(g)$

thus adding term by term

$$f+g \le \max f+ \max g \implies \max(f+g) \le \max f+ \max g $$

Answer 2

More generally, $$\sup(f+g)\leq \sup(f)+\sup(g).$$

Indeed, let $A=\sup(f+g), F=\sup(f)$ and $G=\sup(g)$. Let $\varepsilon>0$. Then, there is $x$ s.t. $$A-\varepsilon\leq f(x)+g(x)\leq F+G.$$ The claim follow.