What is the proof, that $\sum_{n=0}^{\infty} \frac{1}{2^n}\frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}$ is a metric?
108779
How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$
90749
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
153905
If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $
146215
Metric on a Quotient of the Riemann Sphere
141643
Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
111705
Why does $\frac{1+t}{(1-t)^3}=\sum_{n=0}^{\infty}(1+n)^2t^n$
124688
Question about a metric on $\mathbb{R}^2$
132126
Proving $\sum\limits_{n=1}^{\infty } \frac{\mu (n)}n=0$ where $\mu$ is the Möbius function
16143
How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
33859
How to show that a disjoint sum of metric spaces is metrizable?
127034
Let $\rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$, show that $\rho$ is a metric equivalent to $d$
3735
If $\sum_{1}^{\infty}(a_n)^3$ diverges, does $\sum_{1}^{\infty}(a_n)$?
146485
Prove that the product space is a metric space.
151984
Proving the existence of a metric space
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$