Does anybody know how the following equation:
$$|a|+|b| \ge |a+b|$$ changes when a $c$ is added? This way:
$$|a| + |b| + |c| \ge |a+b| + |c| \ge |a+b+c|$$
How should I expand this? What happens when a $c$ is added to the equation?
I'd be grateful for help!
You can prove your version with the extra term involving $c$ by using your first inequality a few times with substitution. First to make things clear re-write the first inequality as $$|x|+|y| \ge |x+y|. \tag{1} $$ Now use (1) with $x=a,y=b$ to get $|a|+|b| \ge |a+b|$ (actually your first ineuality) to which you can add $|c|$ to both sides giving the first of the two inequalities you want, namely $$|a|+|b|+|c| \ge |a+b|+|c|. \tag{2}$$ Now in (1) put $x=(a+b),y=c$ to get $$|(a+b)|+[c| \ge |(a+b)+c|=|a+b+c|, \tag{3}$$ the last equality since we don't need the parentheses on the first term on the right of (3).
Finally putting (2) and (3) together gives your desired inequality.
