= Solution
{c}
The <arctangent pullback metric> is not <complete>. Take $x_n=n$. Since $\arctan n\to\pi/2$, its transformed <sequence> is <Cauchy>, and hence $(x_n)$ is a <Cauchy sequence> for $\rho$. If it converged in $\rho$ to $x\in\mathbb R$, then
$$
|\arctan n-\arctan x|\longrightarrow0
$$
would force $\arctan x=\pi/2$, which is impossible. The missing endpoint of $I$ explains the failure of <completeness>, even though part (i) shows that the <metric topologies> agree.
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