For which metric spaces is Gromov-Hausdorff distance actually achieved?

Unless I am missing something, the distance is achieved for compact metric spaces. Take isometries $i_n :M \to K_n$ and $j_n : N \to K_n$ that give you the Gromov-Hausdorff distance up to $1/n$. Let $K$ be an ultraproduct of the $K_n$ and $i$, $j$ the isometries from $M$, $N$ into $K$ induced by $i_n$, $j_n$. This achieves the distance, I think, when $M$ and $N$ are compact. To see that, given $x$ in $M$ take $y_n$ in $N$ s.t. the distance of $i_n(x) $ to $j_n(y_n)$ is less than the Hausdorff distance from $i_n(M)$ to $j_n(N)$. Let $y$ be the ultralimit of $y_n$ in $N$ (this is where compactness is used). Then the distance from $i(x)$ to $j(y)$ is at most the Gromov-Hausdorff distance from $M$ to $N$.

This is a pretty obvious argument, so I guess it is either written somewhere or is complete nonsense.