Huard’s result on global convergence for a general algorithm (modeled by a point-to-set mapping) [see P. Huard, Math. Program. Study 10, 98- 103 (1979; Zbl 0401.90107)] is generalized in such a way that it may be used also for algorithms which include different point-to-set mappings at each iteration. Moreover, it is shown that this new result is also a generalization of Polak’s and Meyer’s convergence theorems.