Introduction to Mathematical Analysis I - Second Edition

19 (1d) For each x ∈ R , there exists a unique element − x ∈ R such that x +( − x ) = 0. (2a) ( x · y ) · z = x · ( y · z ) for all x , y , z ∈ R . (2b) x · y = y · x for all x , y ∈ R . (2c) There exists a unique element 1 ∈ R such that 1 6 = 0 and x · 1 = x for all x ∈ R . (2d) For each x ∈ R \ { 0 } , there exists a unique element x − 1 ∈ R such that x · ( x − 1 ) = 1. (We also write 1 / x instead of x − 1 .) (2e) x · ( y + z ) = x · y + x · z for all x , y , z ∈ R . We often write xy instead of x · y . In addition to the algebraic axioms above, there is a relation < on R that satisfies the order axioms below: (3a) For all x , y ∈ R , exactly one of the three relations holds: x = y , y < x , or x < y . (3b) For all x , y , z ∈ R , if x < y and y < z , then x < z . (3c) For all x , y , z ∈ R , if x < y , then x + z < y + z . (3d) For all x , y , z ∈ R , if x < y and 0 < z , then xz < yz . We will use the notation x ≤ y to mean x < y or x = y . We may also use the notation x > y to represent y < x and the notation x ≥ y to mean x > y or x = y . A set F together with two operations + and · and a relation < satifying the 13 axioms above is called an ordered field . Thus, the real numbers are an example of an ordered field. Another example of an ordered field is the set of rational numbers Q with the familiar operations and order. The integers Z do not form a field since for an integer m other than 1 or − 1, its reciprocal 1 / m is not an integer and, thus, axiom 2(d) above does not hold. In particular, the set of positive integers N does not form a field either. As mentioned above the real numbers R will be defined as the ordered field which satisfies one additional property described in the next section: the completeness axiom. From these axioms, many familiar properties of R can be derived. Some examples are given in the next proposition. The proof illustrates how the given axioms are used at each step of the derivation. Proposition 1.4.1 For x , y , z ∈ R , the following hold: (a) If x + y = x + z , then y = z ; (b) − ( − x ) = x ; (c) If x 6 = 0 and xy = xz , then y = z ; (d) If x 6 = 0, then 1 / ( 1 / x ) = x ; (e) 0 x = 0 = x 0; (f) − x = ( − 1 ) x ; (g) x ( − z ) = ( − x ) z = − ( xz ) . (h) If x > 0, then − x < 0; if x < 0, then − x > 0; (i) If x < y and z < 0, then xz > yz ; (j) 0 < 1.