Inequality to prove:
$|a+b|\leq |a| + |b|$
$-|a| \leq a \leq |a|$
$-|b| \leq b \leq |b|$
Add 1 and 2 together to get:
- What is the meaning of adding inequalities 1 and 2? How does adding these inequalities rely on the order axioms? Could someone break down this addition into its elementary meaning? I think adding inequalities must be some "short-cut" or "trick" what is the principle behind this?
What I don't understand is how to do this without "adding inequalities". Suppose I use the order axioms and add $-|b|$ to inequality 1. We get:
$-|a|+-|b| \leq a+-|b| \leq |a|+-|b|$
This gives the very left inequality of the final result, but how can we use the order axioms to get the final result?