The problem words:
Let $f: \mathbb{R}\rightarrow \mathbb{R}, x \mapsto \frac{x}{1+\left | x \right |}$ and let $d(x,y):=\left | f(x)-f(y)\right |$ be a metric on $\mathbb{R}$. Show that $x_n\overset{n}{\rightarrow}0\Leftrightarrow d(x_n,x){\rightarrow}0$.
I could show, that usual convergence implies d(x,y) convergence, but I'm lost on the other direction.
Can anyone give me a hint?
The particular form of $f$ does not matter much: all we need to know is that $f$ is continuous and strictly monotone. Continuity gives $$|x_n-x|\to 0 \implies |f(x_n)-f(x)|\to 0$$ for the reverse implication, either use the continuity of $f^{-1}$ (if this fact is available), or argue directly. Namely, given $\epsilon>0$, let $\delta$ be the smaller of the numbers $|f(x\pm \epsilon)-f(x)|$. If $|f(x_n)-f(x)|<\delta$, then $|x_n -x|<\epsilon$ (why? use monotonicity).
