## 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.