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