query: Prove that $||a|-|b|| \leq |a-b|$?
positive_passages.docid: 3820
positive_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 61799
negative_passages.text: Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
negative_passages.docid: 67483
negative_passages.text: Show that $|a-b|<\epsilon \Rightarrow |a|<|b|+\epsilon$
negative_passages.docid: 127296
negative_passages.text: prove that a|b and b|a iff a=+-b
negative_passages.docid: 26766
negative_passages.text: Proving : $A \cap (B-C) = (A \cap B) - (A \cap C)$
negative_passages.docid: 96217
negative_passages.text: If a and b are non-negative real numbers then demonstrate inequality
negative_passages.docid: 12954
negative_passages.text: Prove if $ac>bc$ and $c>0$, then $a>b$
negative_passages.docid: 143979
negative_passages.text: How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 51347
negative_passages.text: Prove that $|a|\leq \max\{|b|,|c|\}$ if $b\leq a \leq c$
negative_passages.docid: 155884
negative_passages.text: What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
negative_passages.docid: 143130
negative_passages.text: Proving that the supremum of $(a,b)$ is $b$.
negative_passages.docid: 31465
negative_passages.text: Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
negative_passages.docid: 80255
negative_passages.text: How to prove that $|a-b|<\epsilon$ implies $|b|-\epsilon<|a|<|b|+\epsilon$?
negative_passages.docid: 59513
negative_passages.text: For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
negative_passages.docid: 122326
negative_passages.text: Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
negative_passages.docid: 77192
negative_passages.text: How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?