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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.


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