To prove that $\frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ is a metric on $\mathbb{C}$
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
105997
Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$
145632
Proving $\frac{1}{\cos^2\alpha}<\frac{\tan\beta-\tan\alpha}{\beta-\alpha}<\frac{1}{\cos^2\beta}$
16774
If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
35291
How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric?
140692
Existence of a complete metric on $(0,1)$
121218
Is this function a metric?
78108
Do these define metrics on $\mathbb{R}$?
117046
Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
61262
Is $\cot(z)-\frac1z$ bounded on a given circle?
23845
How to prove the continuity of the metric function?
117418
Showing that the image of a function is $\mathbb{C}$ if it satisfies a nice functional equation
76490
Proving that $\cos(\arcsin(x))=\sqrt{1-x^2}$
66082
Using $\sqrt{1-t}\leq 1-\frac t2$ to show that $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n}\geq\frac1{2\sqrt n}$
71796
Proving $\rm \frac i 2 \ln\frac {x+i}{x-i}=\arctan x$ .