showing completeness of a metric space $X=\mathbb{R}_{>0}$, $d(a_1,a_2)=|\ln(a_1)-\ln(a_2)|$. 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 $|\ln(a_n)-\ln(a)| \to 0$ $(n \to \infty)$. 

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 $L$.
