In this talk, we shall discuss various notions of convergence of dense graphs, and attempt to come up with a metric space where the notion of "left convergence" of graphs can be identified with Cauchy convergence. This will be an attempt to get a glimpse on the recent line of work on convergence in graphs by Lovász and his co-authors.
Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T., & Vesztergombi, K. 2008). Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 219(6), 1801-1851.