Why $\mathbb R$ is not complete with the metric $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$?

Question

Suppose $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$. Prove that $\mathbb R$ is not complete with this metric.

This is exercise 12 from chapter 1 from Rudin's Functional Analysis.

I tried to find a Cauchy sequence for this metric that does not converge but I couldn't.

@RobinGoodfellow I tried to find a cauchy sequence for this metric that not converge but I couldn't. — ta_frimousse, Oct 12 '14 at 23:16

score 3 · Accepted Answer · answered Oct 13 '14 at 01:00

3

Hint: Try with $x_n=n$. It is a $d$-Cauchy sequence but not $d$-convergent.

answered Oct 13 '14 at 01:00

Hamou

1 Answers1