Question

show that \(\left| |\mathbf{a}|-|\mathbf{b}|\right| \leq |\mathbf{a}-\mathbf{b}|,\quad\forall\:\mathbf{a,b}\in\mathbb{R}^n\)

Answer 1

(a - b)^2 = a^2 - 2ab + b^2 >= a^2 - 2|a||b| + b^2 = |a|^2 - 2|a||b| + |b|^2

so,

(a - b)^2 >= |a|^2 - 2|a||b| + |b|^2

which leads to:

(a - b)^2 >= (|a| - |b|)^2

----> |a - b| >= ||a| - |b||

Answer 2

thanks!