Semi-continuity of metric projections in $\ell_\infty$-direct sums

Let $Y$ be a proximinal subspace of finite codimension of $c_0$. We show that $Y$ is proximinal in $\ell_\infty$ and the metric projection from $\ell_\infty$ onto $Y$ is Hausdorff metric continuous. In particular, this implies that the metric projection from $\ell_\infty$ onto $Y$ is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.