Show that $d(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$ defines a metric on $(0,\infty)$.
82961
Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
140692
Existence of a complete metric on $(0,1)$
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
8913
Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
128598
How to prove this is a metric?
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?
4735
Proof for alternative definition of the derivative
94218
Showing $d(x,y) = \frac{|x-y|}{1+|x-y|}$ is a distance.
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
149827
show that $x\sin\left(\frac{1}{x}\right)$ is uniformly continuous on $(0,1)$
75391
How to prove that a left-invariant metric on a Lie group is smooth?
83159
Defining a metric space
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
146215
Metric on a Quotient of the Riemann Sphere
156727
A step in the proof of Characterization of $W^{1,\infty}$
105997
Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$