Increasing the degree of closed B-spline curves

We consider such sequences of closed B-spline curves the elements of which differ only in their degree; i.e., they share the same control polygon. First, we prove that a sequence obtained in this way converges to the point that represents the arithmetic mean of the common control points (their position vectors) as the degree tends to infinity. Then, we show that this property can be generalized to closed rational B-spline curves in which case the limit is the weighted arithmetic mean of control points.