query: Prove the triangle inequality
positive_passages.docid: 22237
positive_passages.text: Proof of triangle inequality
negative_passages.docid: 140299
negative_passages.text: Inequality based on triangle: $\frac{3}{2}\le\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 106979
negative_passages.text: Prove the inequality and limitation
negative_passages.docid: 111903
negative_passages.text: An inequality for sides of a triangle
negative_passages.docid: 78144
negative_passages.text: How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?
negative_passages.docid: 32963
negative_passages.text: proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
negative_passages.docid: 126632
negative_passages.text: Proving Inequalities with Absolute values
negative_passages.docid: 133428
negative_passages.text: How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$
negative_passages.docid: 114297
negative_passages.text: Extremizing sum and difference of lengths using triangle inequality
negative_passages.docid: 120309
negative_passages.text: Number Theory Inequality Proof
negative_passages.docid: 105268
negative_passages.text: prove inequality through mean value theorem
negative_passages.docid: 29198
negative_passages.text: How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?
negative_passages.docid: 150609
negative_passages.text: Triangle Inequality of the Cartesian product with Max function
negative_passages.docid: 73029
negative_passages.text: If a,b,c are sides of a triangle, prove: $ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \le \sqrt{a} + \sqrt{b} + \sqrt{c} $
negative_passages.docid: 126557
negative_passages.text: Proving the inequality $\sum_{\text{cyc}} \frac{a}{a^2+b^3+c^3}\le\frac1{5abc}$
negative_passages.docid: 15365
negative_passages.text: Proving that the triangle inequality holds for a metric on $\mathbb{C}$
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$