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\| $