Pointwise approximation by a Runge exhaustion
= Pointwise approximation by a Runge exhaustion
To approximate a piecewise constant function pointwise, exhaust each of its separated regions by compact subsets whose finite union has connected complement. Apply polynomial Runge approximation to the locally constant holomorphic function on each stage, with errors tending to zero. Every fixed point eventually lies in all later stages, so the uniform stagewise estimates imply pointwise convergence.