1.8 Completeness of ℝ
Recall the following important definition from calculus, completeness of ℝ.
Definition 1.8.1 A non empty set, S ⊆ ℝ is bounded above (below) if there exists x ∈ ℝ
such that x ≥
s for all s ∈ S. If S is a nonempty set in ℝ which is bounded above,
then a number, l which has the property that l is an upper bound and that every other upper
bound is no smaller than l is called a least upper bound, l.u.b.
. If S is a
nonempty set bounded below, define the greatest lower bound, g.l.b.
Thus g is the g.l.b.
means g is a lower bound for S and it is the largest of all lower
bounds. If S is a nonempty subset of ℝ which is not bounded above, this information is
expressed by saying
∞ and if S is not bounded below,
Every existence theorem in calculus depends on some form of the completeness axiom.
Axiom 1.8.2 (completeness) Every nonempty set of real numbers which is bounded above
has a least upper bound and every nonempty set of real numbers which is bounded below has
a greatest lower bound.
It is this axiom which distinguishes Calculus from Algebra. A fundamental result about sup
and inf is the following.
Proposition 1.8.3 Let S be a nonempty set and suppose sup
exists. Then for every
exists, then for every δ >
Proof: Consider the first claim. If the indicated set equals ∅, then sup
is an upper
bound for S
which is smaller than sup
contrary to the definition of sup
as the least
upper bound. In the second claim, if the indicated set equals
be a lower bound which is larger than inf
contrary to the definition of inf