To show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. Show that $X = (0,1]$ is complete with respect to the metric $e $ where  $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$.
My proof: let $(x_n)$ be Cauchy in $(X,e)$. Let $(t_n) := \frac{1}{(x_n)}$. Then $(t_n)$ is Cauchy in $[1, \infty )$. Hence there exist a $t \in [1, \infty )$ such that $t_n \to t$ and this implies that $x_n \to 1/t$ in $(X,e)$. Thus....
Is the line of proof ok??

I think what you're trying to say is this: if two metric spaces are isomorphic (that is, there exists a set bijection between them that plays well with the two metrics), then if one is complete then so is the other one. This is a true fact (it's a good exercise to prove it).
And you're also saying that the map from $X$ in this problem, to $[1,\infty)$ with the usual Euclidean metric, given by $x\mapsto \frac1x$ is a metric space isomorphism. This is also true, and almost trivial to prove given the definition of $X$'s metric.
