Suppose X is a topological space and Y a proximity space ,
fn L C f (Leader Convergence) iff for each A in X, B in Y, f(A) near B implies eventually fn (A) is near B. L.C. is a generalization of
U. C. (Uniform Convergence). In this paper we study L. C. and various generalizations and prove analogues of the classical results of Arzelà, Dini and others