Abstract. A notion of Cauchy net is introduced into the constructive theory of apartness spaces. It is shown that for a sequence in a metric space this notion is equivalent to the standard metric notion of Cauchy sequence. Applications of this notion are then given, culminating in a generalisation of Bishop’s Lemma on locatedness. 1 Introduction Axioms for a constructive theory of apartness between sets were introduced in [12], where the particular example of a uniform space was discussed in detail. In the present paper we discuss Cauchy and convergent sequences in the framework of that theory. By constructive mathematics we mean mathematics developed with intuitionistic logic