query: Proving the inequality for all positive reals
positive_passages.docid: 17100
positive_passages.text: Prove that $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b} \ge a+b+c$
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 95293
negative_passages.text: Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
negative_passages.docid: 77493
negative_passages.text: Classic $c_r$-inequality in $l_r$ space: $E(|X+Y|^r)\leq c_r[E(|X|^r)+E(|Y|^r)]$
negative_passages.docid: 11983
negative_passages.text: Proof of Clarkson's Inequality
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 143968
negative_passages.text: Proving this inequality by mathematical induction
negative_passages.docid: 126032
negative_passages.text: Proving a certain inequality
negative_passages.docid: 22237
negative_passages.text: Proof of triangle inequality
negative_passages.docid: 27587
negative_passages.text: inequality on the unit disk
negative_passages.docid: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$
negative_passages.docid: 139692
negative_passages.text: Proof of inequality using AM-GM inequality
negative_passages.docid: 44823
negative_passages.text: Prove the general arithmetic-geometric mean inequality
negative_passages.docid: 98002
negative_passages.text: Let $x$ be a positive real number. Inequality problem with $\pi$ and $x$ terms.
negative_passages.docid: 19078
negative_passages.text: Prove an interpolation inequality
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$