One of the difficulties associated with analysis is that it deals with continuous structures, which tend to be not just infinite but uncountable. However, this is not always as much of a difficulty as it appears, and one of the ways of getting round it is discretization: that is, approximating a continous structure by a discrete one. Usually, one proves a result about a sequence of discrete structures that give better and better approximations to the continuous structure, and then one deduces facts about the continuous structure by means of suitable limiting arguments.
A thorough knowledge of basic analysis.
Links to articles about different kinds of discretization arguments
Discretization followed by compactness arguments Quick description ( A useful technique for proving topological statements such as fixed point theorems is to prove discrete analogues and then use compactness to deduce continuous versions from the discrete ones. )
Discretization and differential equations Quick description ( This is a big area, but one method of approximating solutions to differential equations is to replace them by closely related difference equations. Until this Tricki article (or navigation page) is written, this Wikipedia article can serve as an introduction to the basic idea. But the technique can also be used by pure mathematicians: it certainly doesn't always work, but sometimes one can prove the existence of a solution to a partial differential equation by proving rigorously that some kind of limit of solutions of related difference equations does the job. See also the related Wikipedia article on the so-called finite element method. )