Inequality Proof: Max {A+B,C} ≤ Max {A,C} + Max {B,C}

[stanley.st]

Hello,

i've met during problem solving with inequality

$$\max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\}$$

where A,B and C are real numbers. I don't know whether it holds, but I need to prove that.

Thanks for reply...

 

[JCVD]

It holds if A,B,C are nonnegative, but play around with some negative numbers and you'll find a counterexample. (Hint: Does the inequality always hold if max{A+B,C}=A+B, or if max{A+B,C}=C? Pick the case you can't prove easily and search for counterexamples there.)