Start Learning Reals 4 | Construction [dark version]

The construction of the real numbers is built from rational Cauchy sequences: start with all sequences of rational numbers that get arbitrarily close to each other, then group together any two sequences that “should represent the same point” by requiring their term-by-term difference to converge to zero. The key move is to avoid relying on limits inside the rationals (since some Cauchy sequences in Q have no rational limit). Instead, equivalence classes of Cauchy sequences become the new objects—each class corresponds to one real number.

The process begins with the set C of all Cauchy sequences (xn) where every xn is rational. Two sequences a_n and b_n are declared equivalent if their “gap” shrinks to nothing in the sense that the sequence (b_n − a_n) converges to 0. This captures the geometric intuition: if two rational sequences represent the same point on the number line, their terms cluster around that point, so the distance between corresponding terms becomes arbitrarily small. With this definition, equivalence classes are formed; each class contains all Cauchy sequences that represent the same real number.

A real number is then defined as one of these equivalence classes. For example, the rational point 1/3 can be represented by the constant Cauchy sequence 1/3, 1/3, 1/3, …; another representation uses a decimal-style sequence 0.3, 0.33, 0.333, …, which also converges to 1/3. These two sequences land in the same equivalence class because their difference tends to zero, even though the construction is designed to work without explicitly taking limits in Q.

Once the real numbers are defined as equivalence classes, the arithmetic operations are defined by acting term-by-term on representatives. Addition of two classes is defined by adding the corresponding rational terms of the chosen Cauchy sequences; multiplication is defined similarly using rational multiplication. The construction requires a well-definedness check: the result must not depend on which representative sequence is chosen from each equivalence class. The ordering is handled more carefully. Directly comparing all terms would be too strict, since two sequences could straddle the same real number while still converging toward it. Instead, the comparison is made “eventually”: one real number is greater than another if, beyond some index N, the representative terms stay on the correct side (and the discussion refines this idea using a distance-based viewpoint to avoid contradictions when sequences approach the same point from opposite sides).

Finally, the completeness property is emphasized: by construction, every Cauchy sequence of real numbers converges to a real number in this system. That completeness is what distinguishes the reals from the rationals and ensures that limits needed for analysis are always available. The construction closes by noting that this framework sets up the next step toward complex numbers.