Continuity

1  Real numbers

The real numbers consist of: a set R, together with two binary operations, formally functions R× R ® R, written (a,b) |® a+b and (a,b) |® ab; two distinguished elements (i.e. elements of R with definite names) 0 and 1; and a binary relation written a<b. This set-up obeys the usual axioms (associativity etc.).

2  The real numbers as a complete ordered field

Definition 1   A system satisfying the usual axioms of addition and multiplication for numbers is called an (algebraic) field.
Definition 2   A system satisfying the usual axioms of addition and multiplication for numbers, and the usual axioms for ordering, is called an ordered field.

Completeness axiom: Every non-empty bounded subset S of R has a least upper bound supS in R.


Suppose now that F is (just known to be) an ordered field. Then the Completeness axiom makes sense in F, by the following definitions:
Definition 3   For x,y Î F we define x £ y to mean (either x<y or x=y).

Definition 4   If S Í F is a subset then U Î F is called an upper bound for S if s £ U " s Î S.

Definition 5   If S Í F (S ¹ Ø) has an upper bound then we say that S is bounded above.

Then the Completeness axiom makes sense in F, though it may or may not be true in F. We assume it is true for R, so we take R to be a complete ordered field.
Definition 6   A least upper bound for a nonempty subset S of any ordered field F is an element of F such that:
  1. is an upper bound for S, and
  2. If u is any upper bound for S then £ u.

Traditionally we write supS for the least upper bound of S Í R.

We'll assume R contains subsets N, Z, Q with all the usual properties.
Proposition 7   If S Í R has a least upper bound, then it has at most one.

Proof: Suppose and ' are both least upper bounds for S Í R. View as an upper bound and ' as least such, and see from (2) that ' £ . Now interchange the rôles of and ', and get £ '. So ='.

Since by the axiom any non-empty S Í R which is bounded above has a least upper bound, we now see that it is unique. Call it supS (supremum or l.u.b.).

2.1  Remark

If F Í R is finite (non-empty) then F contains a greatest member, and this is supF.

2.2  Remark

More generally (even if S isn't finite) it may be that S contains a greatest element; call it x