prove the following using the triangle inequality

Question

QUESTION: https://i.stack.imgur.com/TB1Cq.jpg

can anyone check my MY ANS:

let a , b, c ∈ R. then

| |a| + b - c| = ||a|| + (b - c)|

| |a| + b - c| ≤ ||a|| + |b - c| ---- by triangle inequality and x=|a| and y= b - c

≤ ||a|| + |b| + |-c| ---- triangle again on |b - c| [and -c = +(-c)]

≤ |a| + |b| + |c| ---- |-c| = |c| and ||a||=|a|

Answer 1

Your answer looks OK

By the triangle inequality: $$|a|+|b|+|c|=||a||+|b|+|-c|\geq||a||+|b-c|\geq||a|+b-c|.$$