Show that $d'$ is a metric
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
104263
How to show that $f'(x)<2f(x)$
104413
Is the exponential of $-d$ a positive definite kernel for a general metric space $(X, d)$?
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
33859
How to show that a disjoint sum of metric spaces is metrizable?
124179
Proof that every metric space is normal
124688
Question about a metric on $\mathbb{R}^2$
46834
show that a distance function is continuous
6415
Distance function is continuous in topology induced by the metric
108779
How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$
128894
About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
157742
Prove that $U(n)$ is a manifold
144353
A closed ball in a metric space is closed
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
72033
Is this proof sufficient to show that a concave function of a metric is also a metric?
77867
Existence of a Riemannian metric inducing a given distance.