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 S_n that converges to S, consider the subset, A = {S_n : S_n ≤ S}, containing all the elements of S_n that are not larger than S.  The supremum of A is S.  Conversely, given an infinite bounded set of reals, A, consider the sequence, S_n, 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 S_N of the sequence) such that S_N exceeds S - ε.  Since the sequence is arranged in ascending order, all S_n where n > N also exceed S - ε.  So lim_n—>∞ S_n = 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.

The webmaster and author of this Math Help site is Graeme McRae.