Does anyone know this inequality $\big\|x-y\big\|^2\geq \big\|x\big\|^2+\big\|y\big\|^2-2\big\|x\big\|\big\|y\big\|$

Question

I came across an inequality that I have not seen before. I can not say if this is correct. But two things interest me:

1. Is the inequality correct?
2. Where does the inequality come from?

This is the inequality:

$$\big\|x-y\big\|^2\geq \big\|x\big\|^2+\big\|y\big\|^2-2\big\|x\big\|\big\|y\big\|$$

Answer 1

Since $\lVert x\rVert=\lVert x-y+y\rVert\leqslant\lVert x-y\rVert+\lVert y\rVert$, you have $\lVert x-y\rVert\geqslant\lVert x\rVert-\lVert y\rVert$ and, by a similar argument, $\lVert x-y\rVert\geqslant\lVert y\rVert-\lVert x\rVert$. Therefore$$\lVert x-y\rVert\geqslant\bigl\lvert\lVert x\rVert-\lVert y\rVert\bigr\rvert$$(the so-called inverse triangle inequality) and so\begin{align}\lVert x-y\rVert^2&\geqslant\bigl\lvert\lVert x\rVert-\lVert y\rVert\bigr\rvert^2\\&=\lVert x\rVert^2+\lVert y\rVert^2-2\lVert x\rVert\lVert y\rVert.\end{align}