Definition of "Upper Bound"
C is an upper bound of a set S means x ≤ C
for all x Î S. The set S is said to be
"bounded above" by C.
A function, f, is said to have a upper bound C if f(x) ≤ C for all x in its domain.
The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if
ε is any positive quantity, however small, there is a member that exceeds M -
ε.
The least upper bound of a function, f, is defined as a quantity M such that
f(x) ≤ M for all x in its domain, but if ε is any positive quantity, however small, there is an
x in the domain such that f(x) exceeds M - ε.
The Completeness Axiom of real numbers states:
Any non-empty set of real numbers that is bounded above has a least upper
bound,
which is a real number not necessarily in the set.
The Completeness Axiom is sometimes also formulated as: All convergent
sequences of real numbers converge to a real number. To see this is
true of sequence Sn that converges to S, consider the subset,
A = {Sn : Sn ≤ S},
containing all the elements of Sn that are not larger than S.
The supremum of A is S. Conversely, given an infinite bounded set of reals,
A, consider the sequence, Sn, of elements of A arranged in ascending
order. Let S be the supremum of A. By the definition of supremum,
given any small positive number, ε, you can find an
element of the set (and thus an element SN of the sequence) such that
SN exceeds S - ε. Since the
sequence is arranged in ascending order, all Sn where n > N also
exceed S - ε. So limn-->∞
Sn = S.
The completeness axiom is used to prove
the Uncountability of Reals
the Intermediate Value Theorem
-- if k is between f(a) and f(b), then there exists a c in [a,b] such that
f(c)=k.
the Bounded Value
Theorem, which, in turn, is used to prove
Rolle's Theorem -- that if f(a)=f(b) then f'(c)=0 for some c in
(a,b) -- which, in turn, is used to prove
the Mean Value Theorem -- that
f'(c) = (f(a)-f(b))/(a-b) for some c in (a,b) -- and
the Fundamental Theorem of Calculus
-- if f is the derivative of F, then the integral from a to b of f(x)dx is
F(b)-F(a)
Internet references
Completeness,
from the Wikipedia
Related pages in this website
Arithmetic Rules -- properties of
equality, addition, multiplication, and in particular the Axioms of Real
Arithmetic, including the completeness axiom.
