Perfectly Competitive Markets as the Limits of Cournot Market

We consider a perfectly competitive, partial equilibrium market for a single homogeneous good with a (bounded) continuum of infinitesimal firms. Cost functions are essentially unrestricted and are allowed to vary smoothly across firms. We introduce a sequence (net) of Cournot markets (each with a finite number of firms) which converge smoothly to the perfectly competitive limit in terms of both the inverse demand functions and the distribution of firm technologies and show that all markets sufficiently far along the sequence have a Cournot equilibrium and all the Cournot equilibria converge to the perfectly competitive equilibrium of the limit market.