How do I proof that this is a metric in R?

Solution 1:

One observation is that if $|\frac{x}{1+|x|}-\frac{y}{1+|y|}|=0$ then $x,y$ has to be both positive or both negative. Thus we can consider two different cases.

Case 1: Both positive. Thus we have $|\frac{x}{1+x}-\frac{y}{1+y}|=0$. In other words, $\frac{x}{1+x}=\frac{y}{1+y}$. Now use derivatives if necesary, because $f(x)=\frac{x}{1+x}$ is one to one, we have $x=y$. Alternatively, suggested by Dan, one could just multiply through to have $$x+xy=y+yx$$ and thus $x=y$. (We are allowed to multiply through since neither $x$ nor $y$ were assumed to be negative.

Case 2: Both negative. This is similar to Case 1.

Solution 2:

If $z=\frac{x}{1+|x|}$ then $1-|z|=\frac{1}{1+|x|}$, so $x=\frac{z}{1-|z|}$.