Recall the following important definition from calculus, completeness of ℝ.
Definition 1.8.1A non empty set, S ⊆ ℝ is bounded above (below) if there exists x ∈ ℝsuch 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 upperbound is no smaller than l is called a least upper bound, l.u.b.
(S)
or often sup
(S)
. If S is anonempty set bounded below, define the greatest lower bound, g.l.b.
(S)
or inf
(S)
similarly.Thus g is the g.l.b.
(S)
means g is a lower bound for S and it is the largest of all lowerbounds. If S is a nonempty subset of ℝ which is not bounded above, this information isexpressed by saying sup
(S)
= +∞ and if S is not bounded below, inf
(S )
= −∞.
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 abovehas a least upper bound and every nonempty set of real numbers which is bounded below hasa greatest lower bound.
It is this axiom which distinguishes Calculus from Algebra. A fundamental result about sup
and inf is the following.
Proposition 1.8.3Let S be a nonempty set and suppose sup
(S)
exists. Then for everyδ > 0,
S ∩ (sup (S )− δ,sup (S )] ⁄= ∅.
If inf
(S)
exists, then for every δ > 0,
S ∩ [inf(S),inf(S)+ δ) ⁄= ∅.
Proof:Consider the first claim. If the indicated set equals ∅, then sup
(S)
− δ is an upper
bound for S which is smaller than sup
(S)
, contrary to the definition of sup
(S)
as the least
upper bound. In the second claim, if the indicated set equals ∅, then inf
(S)
+ δ would
be a lower bound which is larger than inf