query: Double absolute value proof: $||a|-|b||\le |a-b|$
positive_passages.docid: 3820
positive_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 105768
negative_passages.text: Proof involving absolute value and maximums
negative_passages.docid: 15104
negative_passages.text: Show that $\big|\frac{a-b}{1-\overline{a}b}\big|=1$, when either $|a|=1$ or $|b|=1$.
negative_passages.docid: 80255
negative_passages.text: How to prove that $|a-b|<\epsilon$ implies $|b|-\epsilon<|a|<|b|+\epsilon$?
negative_passages.docid: 105164
negative_passages.text: If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
negative_passages.docid: 129480
negative_passages.text: Prove that inf(A+B) = infA + infB
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
negative_passages.docid: 16253
negative_passages.text: Proof that $|\sin(nx)| \le n|\sin(x)|$
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 78736
negative_passages.text: Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
negative_passages.docid: 68705
negative_passages.text: Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
negative_passages.docid: 107774
negative_passages.text: Prove $\|x\|_1\le \sqrt n \|x\|_2$
negative_passages.docid: 111186
negative_passages.text: Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 62172
negative_passages.text: Why does $-b < a < b \iff |a| < b$?
negative_passages.docid: 88982
negative_passages.text: Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
negative_passages.docid: 87928
negative_passages.text: Proof of logical equivalence of biconditional and other proposition