query: Proving inequality $ \frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}\le \frac{|f(x)-h(x)|}{1+|f(x)-h(x)|}+\frac{|h(x)-g(x)|}{1+|h(x)-g(x)|} $
positive_passages.docid: 141643
positive_passages.text: Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
negative_passages.docid: 132431
negative_passages.text: Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 110368
negative_passages.text: Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 39073
negative_passages.text: Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
negative_passages.docid: 65562
negative_passages.text: Proving the inequality $\arctan\frac{\pi}{2}\ge1$
negative_passages.docid: 33097
negative_passages.text: Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$
negative_passages.docid: 64486
negative_passages.text: Show that for non-negative measurable functions $f,g$ with $fg \geq 1$ the inequality $(\int f^p)(\int g^p) \geq 1$ holds.
negative_passages.docid: 137614
negative_passages.text: $\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
negative_passages.docid: 69653
negative_passages.text: Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$
negative_passages.docid: 118613
negative_passages.text: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
negative_passages.docid: 19896
negative_passages.text: Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
negative_passages.docid: 110153
negative_passages.text: How to prove this inequality?$\frac{1}{n+1} + \frac{1}{n+2} + \cdots+\frac{1}{n+n} + \frac{1}{4n} > \ln 2$
negative_passages.docid: 13902
negative_passages.text: Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
negative_passages.docid: 149700
negative_passages.text: Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
negative_passages.docid: 82961
negative_passages.text: Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.