[housecarpenter]

What do you mean by "the use of the Least Upper Bound"?

[griffzhowl]

Probably what Abbott in Understanding Analysis calls the axiom of completeness: every set that is bounded above has a least upper bound.

Making this stipulation distinguishes the reals from the rationals, as e.g. the set of numbers whose square is less than 2 is bounded above by any number whose square is greater than or equal to two, but among the rationals there is no least upper bound: given any rational number whose square is greater than or equal to two we can always find a smaller such rational.

Assuming the axiom of completeness, we define the square root of two as the least upper bound of the set of numbers whose square is less than two

[red_trumpet]

But that is an axiom, not a construction! The point of Dedekind cuts is that they give a construction of the real numbers, and one can prove that this satisfies the Axiom of Least Upper Bounds.

[noqc]

This is basically exactly a dedekind cut.

[davidgrenier]

Where we define the real numbers as the least upper bounds of special sets. There is a bijection between these sets and the set of real numbers which we commonly think of and that bijection is the least upper bound of such sets.

[denotational]

Seconded, the “least upper bound” method for constructing the reals that I know about is… …Dedekind cuts.

[davidgrenier]

I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.