Embedded minimal disks: Proper versus nonproper--global versus local

We construct a sequence of compact embedded minimal disks in a ball in $\mathbf{R}^3$ with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.