Assem, Dupont and Schier introduced the category of rooted cluster algebras, which has as objects pairs (A; ), where A is a cluster algebra (of possibly innite rank) and  a distinguished initial seed of A. We show that, though the category of rooted cluster algebras does not in general admit colimits, every rooted cluster algebra can be written as a directed colimit of rooted cluster algebras of nite rank. Directed colimits of rooted cluster algebras of Dynkin type A have a geometric interpretation as tri- angulations of the closed disc with innitely many marked points on the boundary. They are related to innite discrete cluster categories of type A, respectively the continuous cluster category of type A as introduced by Igusa and Todorov.