定理's post was about how to construct the real numbers (in particular the irrational numbers), so was my post.

All of us kind of know what the real numbers should behave, as we all have studied and used the real numbers.

However, to construct the set of real numbers rigorously/mathematically, we must be super careful. We have to construct the set of real numbers from the bottom up. As 定理 has mentioned, we do this by constructing the integers, rational numbers, and finally the real numbers.

If you have the set of real numbers already constructed for you (or if you don't care how it is constructed), then you know that every Cauchy sequence converges to a real number.

But how do we construct the real numbers? 定理's Dedikind Cut is a way to do this.

And you may also do the construction via Cauchy sequences. To be specific, we may define/construct the set of real numbers to be the set of all Cauchy sequences of rational numbers (modulo the "obvious" equivalence relation). What happens if a Cauchy sequence of rational numbers does not converge to a rational number? Well, this simply says that the sequence represents an irrational number.

Why we can construct the set of real numbers this way? Intuitively speaking, that is because that we know that a Cauchy sequence "should" converge to a real number. How to represent the limit if it is not a rational number? Well, we just use the Cauchy sequence to represent it.

Hope this helped.

It seems that 定理 will explain the Dedikind Cut method carefully in several instalments, which should be more helpful to everyone who is interested in the construction of real numbers.