A dedekind cut comprises two nonempty sets of rationals, L and R, such that each rational appears in exactly one of the two sets, and all the rationals in L (left) are less than all the rationals in R (right). We have cut the line in two, and the cut point becomes the real number.
If x is in R and y > x then y cannot be in L, else y would be less than x. Thus y is in R. Similarly, if x is in L, everything less than x is in L.
If b is an upper bound for L and c is a lower bound for R, c cannot be less than b, and there can't be any gap between, hence b = c. Since each point is suppose to be in just one set, decide arbitrarily that b belongs to L. In this case the real number is the rational b.
One cut is less than another if R1 contains points not in R2. Show this is a partial ordering; in fact it is a linear ordering.
If you have two cuts, and neither is less than the other, the corresponding sets L and R are equal. In other words, the cuts are the same. This is a llittle clearer than the cauchy sequences. Many sequences converge to the same real number, so we had to do all that fiddling with equivalence classes etc, but there is only one dedekind cut for each real number.
Given a rational b, Let L be the points ≤ b, and let R be the points > b. Thus the rationals embed in the reals, and order is preserved.
Two cuts can be added by letting L3 contain sums of points from L1 cross L2, and similrarly for R3. Show that the result is indeed a cut.
Multiplication of nonnegative real numbers is done the same way. The rationals in L3 are the products of the rationals in L1 cross the rationals in L2. Again, show that this is a cut.
Dedekind proved all sorts of nice properties, so that at the end of the day, the cuts form a field, namely the field of real numbers, with the rationals as a dense subfield. We're not going to do that here. Instead, let's map dedekind cuts to cauchy sequences, since we've already done the work over there.
Given a cut, select x in L and y in R. Let z be the average, i.e. the midpoint, and if z is in R, replace y with z, otherwise replace x with z. Repeat this process, and the values of z form a cauchy sequence.
Show that different cuts lead to distinct cauchy sequences. After a while, values of z are held close to their cutpoints, which are separated by some distance d, and that keeps the two sequences apart, so they cannot converge to the same real number. The map is 1-1.
Given a cauchy sequence, and a rational x, subtract x from the terms of the sequence and see if the result is positive or negative. Place x in L or R accordingly. I'll leave the details to you. At the end of the day, cuts and classes of cauchy sequences correspond. Also, the map respects addition and multiplication. It is a ring isomorphism, and since one side is a field, it is a field isomorphism. The cuts define the reals, just as cauchy sequences define the reals.
This was, in many ways, a more direct approach, given the origin of irrational numbers. Euclid showed that sqrt(2) was not rational, and pythagoras showed sqrt(2) exists, as a distance in the plane. A dedekind cut is a natural way to describe sqrt(2). A positive rational x goes into L if x2 < 2, and into R if x2 > 2. Negative values of x go into L. All rationals are covered, and sqrt(2) is defined. This approach seems intuitive for algebraic numbers, and even some transcendentals. For instance, E is the set of rationals x where the integral of 1/t from 1 to x is less than 1.