Victoria University of Wellington
School of Mathematics and Statistics
MATH 313
Topology
2017
Order Properties of the Real Numbers
R = the set of real numbers.
Q = the set of rational numbers.
Z = the set of integers.
Z+ = the set of positive integers (sometimes written N).
• A upper bound of a subset D of R is any c in R such that for all d ∈ D, d ≤ c.
We say that D is bounded above if it has an upper bound.
• A least upper bound of D is an upper bound of D that is less than any other upper
bound of D, so is the smallest upper bound. Thus c is a least upper bound of D iff
(i) c is an upper bound of D; and (ii) if c0 is any upper bound of D, then c ≤ c0 .
The phrase “least upper bound” is often abbreviated to l.u.b., or lub.
A least upper bound is also called a supremum, or sup.
• The order-completeness property of R states that
every non-empty subset of R that is bounded above must have a least upper bound.
• The order-density of Q in R states that between any two distinct real numbers there is
a rational, i.e. if a, b ∈ R and a < b, then there exists a q ∈ Q such that a < q < b.
• The set R − Q of irrationals is also order-dense in R: if a, b ∈ R and a < b, then there
exists an r ∈ R − Q such that a < r < b.
Exercises:
1.
If D has a maximum element m ∈ D, then m is a supremum of D. Conversely, if D has
a supremum that belongs to D, then this supremum is a maximum element.
But in general, a set can have a supremum but no maximum element, e.g. D = the interval
(0, 1).
2.
Define the notions of a lower bound, and greatest lower bound (glb) or infimum (inf)
of a set D ⊆ R. When is D bounded below?
3.
Give a proof that every non-empty subset D of R that is bounded below has a greatest lower
bound.
Hint: apply the order-completeness property to the set of all lower bounds of D.
4.
Prove the Archimedean Property of R: if x ∈ R, there is a positive integer n with x < n.
Hint: apply order-completeness to D = {m ∈ Z+ : m ≤ x}.