triangle inequality of metric d(x,y)

Question

i need help on this problem: how can i prove that the metric d(x,y)=|x-y|/(1+|x-y|) from (ℝ,d) satisfies the triangle inequality?

Greetings

Answer 1

First note that the function $f(x)=\frac{x}{1+x}$ is increasing for positive $x$. You have to prove that $d(a,b)+d(b,c) \ge d(a,c)$ holds for all $a,b,c$. Substituting $|b-c|=n, |a-b|=m$ this is equivalent to $$f(n)+f(m) \ge f(|a-c|)$$ Since $|a-c| \le m+n$ by the triangle inequality (here we use the property of the Euclidean metric not of $d(x,y)$!) and $f$ is increasing, it is sufficient to prove $$f(n)+f(m) \ge f(m+n)$$ But this is just the same as proving $$\frac{n}{1+n}+\frac{m}{1+m} \ge \frac{m+n}{1+m+n}$$ for all non-negative $m,n$. Can you continue from here?