In Introduction to Analysis by Trench, there is a proof by cases for the triangle inequality, stated as follows:
$(a),\ if\ a\geq 0, and\ b\geq 0,\ then\ a+b \geq0\ , So\ |a+b| = a+b= |a|+|b|$ $(b),\ if\ a\leq 0, and\ b\leq 0,\ then\ a+b \leq0\ , So\ |a+b| = -a+(-b)= |a|+|b|$ $(c),\ if\ a\leq 0, and\ b\geq 0,\ then\ a+b = -|a|+|b| , So\ |a+b| = |-|a|+|b||\leq |a|+|b|$ $(d),\ if\ a\geq 0, and\ b\leq 0,\ then\ a+b = |a|-|b| , So\ |a+b| = ||a|-|b||\leq |a|+|b|$
What I wanted to understand is the last two statements (c) and (d). I tried to sketch it as follows so it makes sense if that's true: $||a|-|b||=||b|-|a||\leq ||b|+|a||=|b|+|a|$
Is it the way it is done to show the (c) and (d) are true?