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??
The idea is OK, but should be formalised a bit more.
Let $Y = [1,\infty)$ with the Euclidean metric $d$. Clearly, $Y$ is complete as it is a closed subset of the complete $\mathbb{R}$ in the Euclidean metric.
Then $f(x) = \frac{1}{x}$ from $(X,e)$ to $(Y,d)$.
Then $d(f(x), f(y)) = |f(x) - f(y)| = |\frac{1}{x} - \frac{1}{y}| = e(x,y)$ for all $,y \in X$. So $f$ is an isometry. In particular, it is continuous, both ways, the same formula $g(x) = \frac{1}{x}$ defines an isometry from $(Y,d)$ to $(X,d)$.
So if $(x_n)$ is Cauchy, then $(t_n) = f(x_n)$ is Cauchy in $(Y,d)$, from being an isometry. And $(Y,d)$ is complete, so it has a limit $t$ and continuity of $g$ then garantuees $g(t_n) = (x_n)$ converges to $g(t)$ in $(X,e)$.
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.
You need to fill in some important details, but the basic idea is fine. You might start like this:
Let $\langle x_n:n\in\Bbb N\rangle$ be a Cauchy sequence in $\langle X,e\rangle$. For $n\in\Bbb N$ let $t_n=\frac1{x_n}$. Then by definition for each $\epsilon>0$ there is an $n_\epsilon\in\Bbb N$ such that $|t_m-t_n|<\epsilon$ whenever $n\ge n_\epsilon$. But that means that $\langle t_n:n\in\Bbb N\rangle$ is Cauchy in $\Bbb R$ with the usual metric, which is complete, so $\langle t_n:n\in\Bbb N\rangle$ converges to some $t\in\Bbb R$.
To finish, show that $t\ge 1$, let $x=\frac1t\in X$, and show that $\langle x_n:n\in\Bbb N\rangle$ converges to $x$ in $X$.
