query: Prove: For any real number $a$, $b$ : $|a - b| \ge |a| -|b|$
positive_passages.docid: 128786
positive_passages.text: How to use triangle inequality to establish Reverse triangle inequality
negative_passages.docid: 118981
negative_passages.text: Proving $\sqrt{ab} = \sqrt a\sqrt b$
negative_passages.docid: 118467
negative_passages.text: Prove that $a^2+ab+b^2\ge 0$
negative_passages.docid: 139979
negative_passages.text: Proof there is a rational between any two reals
negative_passages.docid: 72313
negative_passages.text: Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 114043
negative_passages.text: Prove the existence of a real number satisfying a property
negative_passages.docid: 132098
negative_passages.text: Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
negative_passages.docid: 56311
negative_passages.text: Proving $(1+a_{2})^{2}(1+a_{3})^{3}\cdots(1+a_{n})^{n}\ge n^n$ for positive reals $a_2,\ldots, a_n$ whose product is $1$
negative_passages.docid: 56399
negative_passages.text: Proof for $A \cup B = B$ if and only if $A \subset B$
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 138212
negative_passages.text: Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$
negative_passages.docid: 149287
negative_passages.text: Prove that $A,B>0$ and $A_{ij}0$
negative_passages.docid: 143072
negative_passages.text: Proving $|\sin x - \sin y| < |x - y|$
negative_passages.docid: 27281
negative_passages.text: How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
negative_passages.docid: 84946
negative_passages.text: If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
negative_passages.docid: 64108
negative_passages.text: how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?