### A log metric

Posted:

**Tue Jan 08, 2019 11:05 am**Endow $(0, +\infty)$ with the following metric

$$d(x, y) = \left | \log \frac{x}{y} \right |$$

(i) Verify that $d$ is indeed a metric.

(ii) Is the sequence $a_n =\frac{1}{n}$ convergent under this metric? Give a brief explanation.

(iii) Examine if $(0, +\infty)$ is bounded under this metric.

$$d(x, y) = \left | \log \frac{x}{y} \right |$$

(i) Verify that $d$ is indeed a metric.

(ii) Is the sequence $a_n =\frac{1}{n}$ convergent under this metric? Give a brief explanation.

(iii) Examine if $(0, +\infty)$ is bounded under this metric.