How to prove that $\|x-y\| \geq |\|x\|-\|y\||$?
I am only thinking of for the LHS, $\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}$ but not sure how to manipulate that and how to handle the RHS.
How to prove that $\|x-y\| \geq |\|x\|-\|y\||$?
I am only thinking of for the LHS, $\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}$ but not sure how to manipulate that and how to handle the RHS.
It is equivalent to the triangle inequality: $$ \|x\|=\|(x-y)+y\|\le\|x-y\|+\|y\| $$ Then subtract $\|y\|$ from both sides to get $$ \|x\|-\|y\|\le\|x-y\| $$ Similarly, we can show that $$ \|y\|-\|x\|\le\|x-y\| $$ to get $$ |\|x\|-\|y\||\le\|x-y\| $$