[thaumasiotes]

> The construction of the real number system in Hardy's book, using the Dedekind Cut method is overly complicated - the use of the of Least Upper Bound is much simpler and clearer.

A Dedekind cut is a partition of the rational numbers into two sets, A and B, where every number that belongs to A is less than every number that belongs to B.

The cut represents a rational number if B has a least element, and an irrational number if it doesn't. (In full generality, it's a rational number if either (1) A has a greatest element, or (2) B has a least element, but in the case where A has a greatest element, we transfer that element into B, where it's the least element.)

The real number represented by a Dedekind cut is always the least upper bound of A (and the greatest lower bound of B). How does "the use of the Least Upper Bound" differ from Dedekind cuts? They aren't just the same thing in some arcane abstract sense where they both map onto the real numbers - they're the same thing in the most direct sense possible.

(For comparison, my analysis class defined real numbers as Cauchy sequences of rationals. The limit of such a sequence is a real number, but that real number need not be an upper or lower bound to anything.)