TERM 1
Cut
DEFINITION 1
A cut in Q is a pair of subsets A,B of Q such that - A U B = Q,
A ~= 0, B ~= 0, A intersect B = 0 - If a is in A and b is in B
then a < b - A contains no largest element
TERM 2
Real Number
DEFINITION 2
A real number is a cut in Q. R is the class of all real numbers
x = A|B. (i) A|B = {r in Q : r < 1} | {r in Q : r >= 1} (ii) A|B =
{r in Q : r 0 and r^2 >= 2].
TERM 3
Theorem 2
DEFINITION 3
The set R, constructed of Dedekind cuts, is complete in the
sense that it satisfies the least upper bound property: If S is a
non-empty subset of R and is bounded above then in R there
exists a least upper bound for S.
TERM 4
Dedekind Cuts
DEFINITION 4
In mathematics, a Dedekind cut, named after Richard
Dedekind, is a partition of the rational numbers into two non-
empty parts A and B, such that all elements of A are less
than all elements of B and A contains no greatest element.
TERM 5
Cauchy Sequences
DEFINITION 5
A Cauchy sequence is a sequence whose elements become
arbitrarily close to each other as the sequence progresses.
Under the Cauchy condition: For all epsilon>0 there exists N
in IN such that n,m>=N ==>|am - an|
