In the discussion, the focus is on demonstrating that simple functions are dense in the space of bounded Borel functions on a compact space K, using the supremum norm. Participants emphasize the need to show that the sequence of simple functions converges to a bounded Borel function, specifically that the norm difference approaches zero. The initial poster expresses uncertainty about how to begin the proof but acknowledges the goal of establishing density. The conversation highlights the importance of constructing appropriate simple functions to approximate any given bounded Borel function. This foundational concept is crucial in functional analysis and the study of function spaces.