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.
You don't need a construction for a calculus class. If you do need one Cauchy sequence completion is more generalizable and somewhat easier to work with.
I don’t really know what a “calculus” class is since here (the UK) that term isn’t really used for university-level mathematics; we’d usually say “analysis” instead, but I know that “analysis” is a class in the US too, so I don’t know if calculus is closer to what we would do just prior to university (a bit of limits, differentiation, Riemann integrals, a bit of vector calculus).
Virtually every first year UK undergraduate analysis course will start with a construction of the reals via Dedekind cuts, and this is about the level that this book is pitched at.
The original commenter suggested that “least upper bounds” is a simpler approach, and that Hardy’s book is outdated by using Dedekind cuts; it may be that constructing the reals is not something that would be done at “calculus”-level in the US, but clearly the book isn’t aimed at that level.
Dedekind cuts (or Cauchy sequences) are totally standard, and I don’t think it’s fair to criticise their use at all.
In the U.S., there is typically a separation between calculus and real analysis. Though, the amount of difference between the two depends on the university.
In calculus, there is more emphasis on learning how to mechanically manipulate derivatives and integrals and their use in science and engineering. While this includes some instruction on proving results necessary for formally defining derivatives and integrals, it is generally not the primary focus. Meaning, things like limits will be explained and then used to construct derivatives and integrals, but the construction of the reals is less common in this course. Commonly, calculus 1 focuses more on derivatives, 2 on integrals, and 3 on multivariable. However, to be clear, there is a huge variety in what is taught in calculus and how proof based it is. It depends on the department.
Real analysis focuses purely on proving the results used in calculus classes and would include a discussion on the construction of the reals. A typical book for this would be something like Principles of Mathematical Analysis by Rudin.
I'm not writing this because I don't think you don't know what these topics are, but to help explain some of the differences between the U.S. and elsewhere. I've worked at universities both in the U.S. and in Europe and it's always a bit different. As to why or what's better, no idea. But, now you know.
Side note, the U.S. also has a separate degree for math education, which I've not seen elsewhere. No idea why, but it also surprised me when I found out.
There's a POV that learning math and learning how to teach math effectively are two orthogonal things.
If one only took the method of teaching that is most common in US university lecture halls, and applied it to a small class of pre-teens or teenagers, it probably wouldn't be very effective.
I went to a University of California school which had 3 calculus tracks - one for life/social sciences students (eg biology, econ), one for physical sciences (chemistry, physics, math, ...), and an honors track.
High school went up through what we call Algebra II. Calculus is an Advanced Placement (AP) course that most students don't take.
I took physical sciences calc + multivariate calc (1 year including summer), an intro to proofs and set theory course, and then finally a rigorous construction of reals was taught in our upper division real analysis course. So somewhere in my second year as a math major. Though I had already researched the constructions myself out of curiosity.
Apart from the material being extraneous for anyone outside the major, I think they were in a sense trying to be more rigorous by first requiring set theory which included constructions of the integer and rational number systems.
