Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.3 COMPLETENESS IN AN ORDERED FIELD 315

we are willing to assume that any real number falls between two consecu-
tive integers. We assume the latter property for now, justifying it rigorously
in Chapter 10, where we study both the integers and the real numbers in
greater detail.

THEOREM 2

If a and bare real numbers with a < b, then there exists a rational number q such

that a < q < b.

Proof Given a, b E R with a < b, note that b - a > 0 so that, by Corol-
lary 2, there exists a positive integer n such that l/n < b - a. By the
assumed property of integers, there must exist an integer m such that
m - 1 I an < m, so that (m - l)/n 2 a < mln. Hence we have that the
rational number m/n is greater than a. Furthermore, since (m - l)/n 5 a,
we have mln I a + lln. But a + l/n < b so we conclude m/n < b. Let-
ting q = m/n, we note that q is rational and a < q < b, as desired.

THE INTERMEDIATE VALUE THEOREM

The following theorem is familiar to every student of elementary calculus.

T H E 0 R E M 3 (Intermediate Value Theorem)

If f is continuous on the closed and bounded interval [a, b], with f(a) < f(b) (so

that a # b), and if yo is any real number with f(a) < yo < f(b), then there exists

xo E (a, b) such that f(xo) = yo. y

This theorem states that the graph of a function continuous on a closed
and bounded interval [a, b] must pass through every horizontal line y =
yo, where f (a) < yo < f (b), as opposed to possibly "jumping over" any such
line. Perhaps more than any other property of continuity, and surely more
than the formal definition, this result corresponds to our intuitive under-
standing of a continuous function as one whose graph has no "breaks" and
no "missing points," as shown in Figure 9.3. (You may also want to review

Figure 9.3 Graphic view of the intermediate value theorem.
Since g is continuous on [a, b], and g(a) < yo < g(b), the
line y = yo must intersect the graph of g.