# A simple Inequality: $\frac{a+b}{\max\{a',b'\}}\leq\frac{a}{a'}+\frac{b}{b'}$?

We know $a \geq a' \geq 0$ and $b\geq b' \geq 0$ . How we can prove:

# $\frac{a+b}{\max\{a',b'\}}\leq\frac{a}{a'}+\frac{b}{b'}$

• Do it by cases: either the $\max$ is $a'$ or it is $b'$. Discuss each case separately. Remember that if $a'=\max \left(\left\{a'b'\right\}\right)$, then $b'\leq a'$ and similarly if $b$ is the $\max$. – Git Gud Nov 13 '13 at 14:52
• You're absolutely right. It was really simple. :) – 2012User Nov 13 '13 at 14:56

Hint: Assuming that $a',b'>0$, use the fact that $\max\{a',b'\} \ge a',\ \max\{a',b'\}\ge b'$.