1.5. THE CONSTRUCTION OF THE REAL NUMBERS 9 (h) For b ∈ R, the infinite closed interval (−∞,b] is defined by (−∞,b] = {x ∈ R | x ≤ b}. (i) R = (−∞, ∞). Definition 1.4.2. If x ∈ R, a neighborhood of x is an open interval con- taining x. In many instances, it is helpful to use symmetric neighborhoods. That is, if x ∈ R, a symmetric neighborhood of x is an interval of the form (x − ε, x + ε), where ε 0. These intervals, and their counterparts in other spaces, are used exten- sively throughout analysis. Exercise 1.4.3. Suppose that I is a subset of R. Show that I is an interval if and only if for all a, b ∈ I, with a ≤ b, the closed interval [a, b] ⊆ I. The notion of interval is valid in any ordered field, and we will occasion- ally find this useful. We end this section with a theorem about intervals in R, which is called the Nested Intervals Theorem. Theorem 1.4.4 (Nested Intervals Theorem). Let ([an,bn])n∈N be a nested sequence of closed bounded intervals in R. That is, for any n we have [an+1,bn+1] ⊆ [an,bn], or equivalently, an ≤ an+1 ≤ bn+1 ≤ bn for all n. Then n∈N [an,bn] = ∅. Proof. Let A = {an | n ∈ N}. Then A is bounded above by b1. If a = lubA, then a ∈ n∈N [an,bn]. The nested intervals property is actually not exclusive to the real num- bers. In fact, it is really a theorem about a sequence of nested compact sets in a metric space. This result will be proved in the next chapter. There is often some confusion about the relationship between the Nested Interval Theorem in R and the least upper bound property. Although our proof in R involves the least upper bound property, it can be done in alternate ways which involve sequential compactness. 1.5. The Construction of the Real Numbers We are now ready to proceed with the construction of the real numbers from the rational numbers using the fact that the rational numbers are the field of fractions of Z. We have already defined R as an ordered field in which the least upper bound property holds. We now proceed to build such a field starting from Q. Recall that the absolute value on Q is defined as |a| = a if a ≥ 0, −a if a 0.

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