query: Show $(|a|+|b|)^p \leq |a|^p+|b|^p$ for $0\leq p \leq 1$
positive_passages.docid: 3109
positive_passages.text: Prove that $(p+q)^m \leq p^m+q^m$
negative_passages.docid: 66261
negative_passages.text: Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 67483
negative_passages.text: Show that $|a-b|<\epsilon \Rightarrow |a|<|b|+\epsilon$
negative_passages.docid: 149287
negative_passages.text: Prove that $A,B>0$ and $A_{ij}0$
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
negative_passages.docid: 15232
negative_passages.text: Proving an inequality involving real powers
negative_passages.docid: 49217
negative_passages.text: How do I show that $a^p + b^p > (a + b)^p$ for $a,b>0$ and $0<p<1$?
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: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 79165
negative_passages.text: How to show that $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
negative_passages.docid: 46831
negative_passages.text: Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 52204
negative_passages.text: Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$