I want to prove this statment:
Let $f:[a,b] \to \mathbb{R}$ be a bounded function.
Prove that if $f$ is integrable in $[a,b]$ then $|f|$ is also integrable in $[a,b]$ - HINT: first prove that if $M_f=\sup(f(x):x \in [a,b])$ and $m_f=\inf(f(x):x \in [a,b])$, then $M_{|f|}-m_{|f|} \le M_f-m_f$
Unfortunately I have been trying to prove the hint for over than 3 hours now with no luck.
Any help will be amazing!!
Thanks!