How to prove triangle inequality for the metric $d(x,y)=\frac{2|x-y|}{\sqrt{\left(1+|x|^2\right)}+\sqrt{\left(1+|y|^2\right)}}$
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
14952
Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
32963
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
68986
Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
141643
Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
18016
Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
17264
Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
69653
Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$
62462
Problem with inequality: $ \left| \sqrt{2}-\frac{p}{q} \right| > \frac{1}{3q^2}$
27272
Triangle Inequality with Complex Numbers
156513
Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
41558
Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
26606
Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$
128786
How to use triangle inequality to establish Reverse triangle inequality
128598
How to prove this is a metric?
29679
Prove: $|x-y|\leq |x|+|y|$