TITLE: Is a space metric on the positive real numbers not complete?
QUESTION [5 upvotes]: Say we have a metric space $(\mathbb{R}^+, d)$
where the distance function is 
$d(x,y) = |x - y| + | 1/x - 1/y |$
Then I argue that this metric space is not complete: If we look at the Cauchy sequence $1/x$, which is contained in the metric space, we see that the limit of the sequence $\lim_{x\to \infty} \frac{1}{x} = 0$ is not in the metric space. Hence, the metric space is not complete.
Am I doing something wrong or is this a valid argument/proof?

REPLY [11 votes]: Milo Brandt has shown what is wrong with your argument. In fact $\langle\Bbb R^+,d\rangle$ is complete. Here’s a hint outlining how you might prove that.
HINT: Suppose that $\sigma=\langle x_n:n\in\Bbb N\rangle$ in $\langle\Bbb R^+,d\rangle$ is $d$-Cauchy. 

Show that $\sigma$ is Cauchy in the Euclidean metric, and conclude that $\sigma$ converges to some $x\in\Bbb R$ in the Euclidean metric.  
Show that $x\ge 0$.  
Show that if $x$ were $0$, $\sigma$ wouldn’t be Cauchy after all.  
Show that if $x>0$, $\sigma$ converges to $x$ in the metric $d$ as well as in the Euclidean metric.  
Put the pieces together to conclude that $\langle\Bbb R^+,d\rangle$ is complete.