A point x in a metric space is called a Delta-limit of a sequence x_n if for any other point y, d(x_n,y)>d(x_n,x)+o(1).Surprisingly, this is an equivalent definition of weak limit in Hilbert spaces and l^p, but in many other Banach spaces it is not. Similarly to weak convergence, a bounded sequence has a convergent subsequence provided that the space is asymptotically complete. Asymptotic completeness is a property vaguely similar to reflexivity, and like reflexivity it is satisfied by uniformly convex Banach spaces. This remains true also in metric spaces under a suitable generalization. Delta-limits of sequences are also their asymptotic centers, which implies that Delta-convergence is a natural mode of convergence for iterations of non-expansive maps.