Proving metric( the triangle inequality)
14799
How to show that the spherical metric satisfies the triangle inequality?
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
114047
Metric $d(x,y)=\frac{|x-y|}{\sqrt{1+x^2}\sqrt{1+y^2}}$ on $\mathbb{R}$
68986
Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
83159
Defining a metric space
151984
Proving the existence of a metric space
128598
How to prove this is a metric?
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
3751
Proving a distance between molecules defines a metric space.
65562
Proving the inequality $\arctan\frac{\pi}{2}\ge1$
139692
Proof of inequality using AM-GM inequality
107321
How to show convexity of a ball in metric space?
108779
How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$
146485
Prove that the product space is a metric space.
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
141643
Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
11094
How to prove that the French railways metric is a metric