Bounded interval/Continuous function/Equidistant lower integral/Lower integral/Exercise

Let ${}I$ be a bounded interval and let ${}f\colon I\rightarrow \mathbb {R}$ denote a continuous function which is bounded from below. Suppose that the supremum over all staircase integrals for the equidistant lower staircase functions exists. Show that then the supremum for all staircase integrals for lower staircase functions (that is, the lower integral)

exists and coincides with the supremum first mentioned.

Create a solution