How could we show that the metric space
$X=\{0\} \cup \{\frac{1}{n}:n \in \mathbb{Z} - \{0\}\}$ with the metric it inherits as a subset of $\mathbb{R}$ is complete?
Thoughts
Complete metric spaces are those in which all Cauchy sequences converge to a point within the space. For any Cauchy sequence $(x_n)$ in the space, $|x_n|<1$ and so the sequence is bounded; bounded Cauchy sequences in $\mathbb{R}$ converge in $\mathbb{R}$ and so the limit, $x$ say, lies in $\mathbb{R}$.
Suppose that $x \notin X$. I imagine this leads to a contradiction but I can't see what it is.
Any help would be appreciated. Regards, MM.