query: I cannot figure out how to prove $||x| - |y|| \leq |x - y|$
positive_passages.docid: 3820
positive_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 31846
negative_passages.text: Does $\neg(x > y)$ imply that $y \geq x$?
negative_passages.docid: 33543
negative_passages.text: is "y1 < x < y2" an inequality or inequalities?
negative_passages.docid: 53123
negative_passages.text: Trying to derive a contradiction with this simple inequality,
negative_passages.docid: 104125
negative_passages.text: How prove this inequality??
negative_passages.docid: 109778
negative_passages.text: Proving $(x-c)|(p(x)-p(c))$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 28822
negative_passages.text: $|x| - |y| \leq |x-y|?$
negative_passages.docid: 137614
negative_passages.text: $\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 105316
negative_passages.text: Prove : $|x - y| \leqslant \epsilon$ $\forall \epsilon > 0$ iff $x = y$
negative_passages.docid: 45370
negative_passages.text: Is it true that $|d(x, z) - d(y, z)| \leq d(x, y)$ and how do I prove it?
negative_passages.docid: 103536
negative_passages.text: How to prove $|e^{i\theta} - 1| \leq |\theta|$?
negative_passages.docid: 97584
negative_passages.text: How to prove this property using convexity?
negative_passages.docid: 79686
negative_passages.text: How can I prove these statements?
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 64096
negative_passages.text: Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.