A log metric

General Topology
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Riemann
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Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

A log metric

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Post by Riemann »

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.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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