showing completeness of a metric space

Question

$X=\mathbb{R}_{>0}$, $d(a_1,a_2)=|\ln(a_1)-\ln(a_2)|$. I have already proven that $(X,d)$ is a metric space, but I have some problems showing the completeness.

Let $(a_n)_{n\in\mathbb{N}}$ a Cauchy series in $X$, which means $|\ln(a_n)-\ln(a_m)| \to 0$ for $(n,m \to \infty)$.
Now I don't really know how to show that there exists an $a\in X$ such that $|\ln(a_n)-\ln(a)| \to 0$ $(n \to \infty)$.

score 1 · Accepted Answer · answered Oct 17 '14 at 19:04

1

HINT: If $\langle a_n:n\in\Bbb N\rangle$ is a $d$-Cauchy sequence, then $\langle\ln a_n:n\in\Bbb N\rangle$ is a Cauchy sequence in the usual metric, so it converges to some limit $L$. The limit of your sequence is related to $L$.

answered Oct 17 '14 at 19:04

Brian M. Scott

616,228

So I know that there is an L such that $lim |ln(a_n) - L| = lim |ln(a_n/exp(L))| = 0$ which means that $lim a_n/exp(L) = 1$, or $lim a_n = e^L$ for $n \to \infty$. Right? – Val Oct 17 '14 at 19:32
@Val: Yes, that's right. – Brian M. Scott Oct 17 '14 at 19:37

showing completeness of a metric space

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