$\sup\{(f+g)\}\leq \sup\{f\}+\sup \{g\}$

Question

In the midst of a proof I am reading the following lines are present:

For $I\subseteq [a,b]$ and $f,g: [a,b]\to \mathbb{R}$ be integrable on $[a,b]$:

$\inf\{(f+g)(x):x\in I\}\geq \inf\{f(x):x\in I\}+\inf\{g(x):x\in I\}$

and

$\sup\{(f+g)(x):x\in I\}\leq \sup\{f(x):x\in I\}+\sup\{g(x):x\in I\}$.

Why are they true? Some variation of the triangle inequality?

The definitions of the infimum and supremum? I am familiar with their definitions. — Sujaan Kunalan, Mar 30 '14 at 06:33
This is going to follow directly from the definitions! Work it out. — Euler....IS_ALIVE, Mar 30 '14 at 06:34
As an intermediate step, try showing that $(f+g)(x) \le \sup f + \sup g$ for every $x\in I$. — Greg Martin, Mar 30 '14 at 06:42

score 2 · Accepted Answer · edited Mar 30 '14 at 07:04

2

Spelled out: If $a\ge f(x)$ for all $x\in I$ and $b\ge g(x)$ for all $x\in I$ then $a+b\ge f(x)+g(x)$ for all $x\in I$.

edited Mar 30 '14 at 07:04

Rudy the Reindeer

answered Mar 30 '14 at 06:39

Hagen von Eitzen

I see. Thank you. I don't know why that wasn't apparent to me before. – Sujaan Kunalan Mar 30 '14 at 06:56

1 Answers1