For $x,y \in \mathbb R$, let $d(x,y) = |x-y|/(1+|x-y|)$. Determine whether $d$ defines a metric on $\mathbb R$.
94218
Showing $d(x,y) = \frac{|x-y|}{1+|x-y|}$ is a distance.
146967
Understanding equivalent metric spaces
49240
$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$
19599
$ d(x_n,y_n) \to d(x,y)$ if $x_n\to x$ and $y_n \to y$
69528
Metrizability of quotient spaces of metric spaces
65809
Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
146215
Metric on a Quotient of the Riemann Sphere
4538
Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?
108779
How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$
6415
Distance function is continuous in topology induced by the metric
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
140692
Existence of a complete metric on $(0,1)$
127034
Let $\rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$, show that $\rho$ is a metric equivalent to $d$
140124
The distance function on a metric space
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$