query: Prove: $|x|-|y|\leq |x-y|$
positive_passages.docid: 28822
positive_passages.text: $|x| - |y| \leq |x-y|?$
negative_passages.docid: 88328
negative_passages.text: Proof of $1+x\leq e^x$ for all x?
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 20863
negative_passages.text: how do I prove this inequallity?
negative_passages.docid: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$
negative_passages.docid: 105268
negative_passages.text: prove inequality through mean value theorem
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 33543
negative_passages.text: is "y1 < x < y2" an inequality or inequalities?
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 40930
negative_passages.text: Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$
negative_passages.docid: 96133
negative_passages.text: Prove that $ m_{a} \geq m_{g} \geq m_{h} $ using strict inequalities unless $ a = b $.
negative_passages.docid: 102478
negative_passages.text: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 18016
negative_passages.text: Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
negative_passages.docid: 50273
negative_passages.text: How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?