query: How to prove $|\sqrt{a_n}-\sqrt\ell|\le \sqrt{|a_n-\ell|}$
positive_passages.docid: 23264
positive_passages.text: Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $
negative_passages.docid: 150565
negative_passages.text: Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
negative_passages.docid: 118559
negative_passages.text: Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
negative_passages.docid: 129890
negative_passages.text: Proving inequality $|\sin(n\theta) |\le n\sin\theta$
negative_passages.docid: 8814
negative_passages.text: Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy
negative_passages.docid: 117688
negative_passages.text: Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
negative_passages.docid: 3820
negative_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 108429
negative_passages.text: Show that if $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 156766
negative_passages.text: Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 110826
negative_passages.text: prove that : $ i = \sqrt {-1}\ $
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 103536
negative_passages.text: How to prove $|e^{i\theta} - 1| \leq |\theta|$?