 Solution

For $k>0$, ${\int }_k^{\infty }{x}^{-2}dx=1/k$. Thus the unnormalized integral is $1/k$ and the multiplicative normalization is $k$:Although this proposal does not cover the whole Cauchy distribution probability support, it covers the entire nonzero integrand $f\theta$, which is the required support condition for importance sampling.