TITLE: The continuous as the limit of the discrete
QUESTION [6 upvotes]: Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactiﬁcations to view the continuous as the limit of the discrete; for instance, it is possible to compactify the sequence Z/2Z, Z/3Z, Z/4Z, etc. of cyclic groups, so that their limit is the circle group T = R/Z.".
Could you give me a point of start to understand what idea of compactification is being used there? Where could I find an sketch of proof for that fact?

REPLY [2 votes]: I haven't looked at the link, but it seems likely that the author of said link is discussing
convergence in the Hausdorff topology.  In this context the idea of taking limits is due to Gromov, I believe.  (There is a wikipedia entery on Gromov--Hausdorff convergence which seems to be
the relevant one.)  It is a common technique in parts of geometric topology and geometric group theory.  
If you ask another question on Gromov--Hausdorff convergence, I'm sure it would draw the attention of the (at least) several experts on the topic who I know read MO.