Solution
= Solution
A <complete metric space> is one in which every <Cauchy sequence> converges. On $I=(0,1]$ define
$$
d(x,y)=|x^{-1}-y^{-1}|.
$$
The map $x\mapsto1/x$ is an <isometry> from $(I,d)$ onto the closed subset $[1,\infty)$ of $\mathbb R$, so $d$ is a metric and is complete. That map and its inverse are continuous in the ordinary Euclidean metrics, so it is also a <homeomorphism>. Therefore $d$ induces exactly the usual subspace topology on $I$.