9.3 The Real Field
Remark 9.2.8 C with usual + and • is a field. But it is not an ordered field.
If x — i then i^2 = — 1 y 0, hence property (d) does not hold.Definition 9.2.9 Let F (with +, •) and F' (with ©, ®) be two fields. We say
F is a subfield of F' if F C F' and two operations (B and © when restricted to
F are + and •, respectively. That is, ifx,y € F => x®y — x + y, xQy = x-y.
Then, we have OF — OF', and IF = IF'-
Moreover, if F (with -<) and F' with (with -<') are ordered fields, then we say
F is an ordered subfield of F' if F is a subfield of F' and for Vx € F with(^0) F -< x => (^0) F< <' x.
9.3 The Real Field
Theorem 9.3.1 (Existence & Uniqueness) There is an ordered field R
with lub property 3 Q is an ordered subfield of R. Moreover if R' is another
such ordered field, then K and M! are "isomorphic": 3 a function
i) <j) is 1-1 and onto,
ii) Vx, j/6R,
Hi) Vz, € K with x y 0, we have (j)(x) >- 0).
Theorem 9.3.2 (ARCHIMEDEAN PROPERTY)
x, y £ R and x y 0 => 3n G N (depending on x and y) 3 nx >- y.
Proof. Suppose 3x,y e R with x >- 0 for which claim is not true. Then,
Vn £ N, nx •< y. Let A = {nx : n £ N}. A is bounded above (by y).
a = sup A € R, since R has lub property, x y 0 => a — x < a, so a — a: is not
an upper bound for A.
Therefore, 3m £ N 3 (a — x) ~< mx => a -< (m + l)x. Contradiction
(a = sup A). D
Theorem 9.3.3 (Q is dense in R)
Vx, y £ R with x -< y, 3p£Q3x~<p<y.
Proof, x, y£R, x <y ^ y — x y 0
(By Theorem 9.3.2) 3n £ N 3 n(y - x) y 1 => ny y 1 + nx.
3mi £ N 3 mi >- nx f- (y = nx, x = 1) in Theorem 9.3.2.
Let A = {m £ Z : nx -< m}. A ^ 0, because mi £ A. A is bounded below. So
A has a smallest element mo, then nx -< mo =>• (mo — 1) •< nx.
If not, nx -< mo — 1, but mo is the smallest element: Contradiction.
=> (mo — 1) •< nx -< mo =• nx -< mo ^ nx + 1 -< ny => x -< ^^ -< j/. Let
P = = 6 Q. •