Let $f:[a,b]\to R$ be a bounded and integrable function.
Prove that if $M_f=sup\{f(x)|x\in [a,b]\}$ and $m_f=inf\{f(x)|x\in [a,b]\}$ then $M_{|f|} - m_{|f|} \leq M_f - m_f$.
I tried more than a few attempts, the closest I got was:
$M_f \geq m_f$ and $M_{|f|} \geq M_f$ and also $m_{|f|} \geq m_f$
therefore $M_f - m_f \geq m_f - m_{|f|}$
But I couldn't find a way to plug the $M_{|f|}$ in a way that makes sense.
Can someone help me solve this using this method (supremums and infimums)? It seems pretty trivial
