query: Is $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$ when $a,b \geq 0 $ and $a,b \in \mathbb{R}$?
positive_passages.docid: 25623
positive_passages.text: Does $\sqrt{a+b} \le \sqrt a + \sqrt b$ hold for all positive real numbers a and b?
negative_passages.docid: 139503
negative_passages.text: How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
negative_passages.docid: 73696
negative_passages.text: Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$
negative_passages.docid: 28499
negative_passages.text: Inequality.$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq 3$
negative_passages.docid: 68838
negative_passages.text: Why $\sqrt[3]{3}\not\in \mathbb{Q}(\sqrt[3]{2})$?
negative_passages.docid: 85977
negative_passages.text: Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
negative_passages.docid: 53660
negative_passages.text: Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 28218
negative_passages.text: Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
negative_passages.docid: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
negative_passages.docid: 138944
negative_passages.text: Is $\mathbb Z[\sqrt{-3}]$ Euclidean under some other norm?
negative_passages.docid: 95643
negative_passages.text: Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$
negative_passages.docid: 8027
negative_passages.text: How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$?
negative_passages.docid: 96630
negative_passages.text: Let a and b be real numbers such that 0  \sqrt{ab} > \frac{2ab}{a + b}$
negative_passages.docid: 14221
negative_passages.text: How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
negative_passages.docid: 80772
negative_passages.text: What happens when square root is performed in inequalities?
negative_passages.docid: 87582
negative_passages.text: Some inequalities between 1- norm, 2- norm and infinity-norm: $\|x\|_2\leq\sqrt{\|x\|_1\| x\|_\infty}\leq\frac{1+\sqrt{n}}{2}\|x\|_2$