A SEQUENTIALLY CONVEX HULL DAVID OATES Let U denote the closed unit ball in the Banach space C(S) consisting of all continuous real-valued functions on a totally disconnected compact Hausdorff space S, with the uniform norm. It is well known [1, 2] that although U is not compact it still satisfies the conclusion of the Krein-Milman theorem by being the closed convex hull of its extreme points. In this paper a short proof is given of the stronger property that U is the sequentially-convex hull of its extreme points. That is to say, for every function h in U t there exist sequences (a B ) of positive reals and (e n ) of extreme points of U with This shows that U satisfies the Choquet-type property that each point is the centroid of a probability measure supported by the extreme points of U. Let [f,g] denote the order interval of all continuous functions h on S with f^h4:g, and let ext[/,g] denote the set of extreme points of [f,g]. THEOREM. Let S be a totally disconnected compact Hausdorff space, and let / ^ g be continuous real-valued functions on S with g(x)—f{x) > 0 for all x in S. Then = S-COQ\t[f,g]. Proof It is sufficient to prove the theorem for an interval [—/,/] where f{x) > 0 for all x in S. We show first that for any heC(S) with —/< h ^f there are extreme points e' and e" of [—/,/] with \Kx)-\e'{x)-\e"{x)\^\j{x) for all x in S. (•) Since S is totally disconnected, there is a partition {A,B,C} of S into open and closed sets refining the open covering {U, V, W), where U = {xeS:h(x) > \J{x)}, V={xeS:h(x) < -£/(*)} and W = {xeS:\h(x)\ < ±/(JC)}. The functions e' = (2x A — l)/and e" = (1 — 2^ B )/are extreme points of [—/,/]. For xeA, \h(x)-\e'(x)-\e"(x)\ = \Kx)-iKx)\ ^ ±/O), and similarly for xeB. For xeC, \h(x)-\e'(x)-\e"(x)\ = \h(x)\ ^ §./(*), proving (•). Received 16 November 1989. 1980 Mathematics Subject Classification 46E15. Bull. London Math. Soc. 22 (1990) 467^68