Cutting corners preserves Lipschitz continuity

In this note we prove that the corner cutting procedure preserves continuity properties, i.e., a sequence of polygons obtained in this way belongs to the Lipschitz class of the same constant and exponent. As a consequence this also holds for all functions or curves obtained as the limit of this procedure, such as the Bernstein polynomials, Bézier and spline parametric curves, etc.