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		excursion into elementary real analysis to introduce some tools that help in formalizing these density notions.

		§7.2 Some Background

		Real Numbers, Supremums, and Infimums

		Here we review the small amount of real analysis required for this chapter. For more details on the following, see almost any book on real analysis, for example, Dudley [56] or Rudin [162].

		Recall from Chapter 2 that R denotes the set of real numbers. In this chapter, let a, b, and c range over R. Recall that:

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		Also recall that, for natural numbers x and z:

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		Let X be a nonempty set of real numbers. We define inf(X) as follows. If there is a w such that, for all , , then inf(X) is the largest such w; otherwise, . For example, , but note that . For another example, We define . Equivalently, if there is a y such that, for all , then sup(X) is the smallest such y; otherwise, . Suppose that a0, a1, . . . is a sequence of real numbers. Define:

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		For example, for each n, let and . Then is undefined, but lim and lim . We note that, for any sequence a0, al, . . . ,