Dedekind Cuts vs Cauchy Sequences: Two Constructions of ℝ, Same Number Line

If you have looked at the Dedekind-cut visualiser and at Is \sqrt{2} a real number or just a symbol?, you already know the two main ways of building \mathbb{R} from \mathbb{Q}. But the question you probably did not ask out loud is: why two? If one construction works, why does every analysis textbook mention a second one? And if they are different, how can both of them be called "the real numbers"?

The short answer is that they are different descriptions of the same object. Dedekind cuts and Cauchy sequences use completely different raw material — subsets of \mathbb{Q} versus sequences of \mathbb{Q} — but the number system each one spits out satisfies the same axioms, with the same order, the same arithmetic, and the same limits. A theorem about uniqueness of complete ordered fields says these two systems must be isomorphic. You are free to use whichever description feels more natural for the problem in front of you.

The two recipes, side by side

Both constructions start from \mathbb{Q}, the rationals. Both want to end at \mathbb{R}, a larger number system in which every bounded set has a supremum, every Cauchy sequence converges, and every continuous function that changes sign has a root. They differ in what they declare "a real number" to be.

Dedekind's recipe (1872). A real number is a partition of \mathbb{Q} into a pair of sets (L, U) such that

1. L and U are both non-empty;
2. every element of L is strictly less than every element of U;
3. L has no largest element.

You already saw this in the widget: drag a cut along \mathbb{Q}, and whatever lands left of the cut becomes L and whatever lands right becomes U. The cut is the real number. For \sqrt{2}:

Cantor's recipe (1872). A real number is an equivalence class of Cauchy sequences of rationals, where two sequences (a_n) and (b_n) are equivalent if the sequence (a_n - b_n) converges to 0.

A Cauchy sequence of rationals is a sequence whose terms eventually stay as close together as you want — for every positive rational \varepsilon, there is an index N beyond which |a_m - a_n| < \varepsilon for all m, n > N. The terms "bunch up." For \sqrt{2}, the canonical representative is

but 1.5, 1.42, 1.415, 1.4143, \ldots and the Newton-iteration sequence 1, 1.5, 1.41\overline{6}, 1.41421568\ldots are all in the same equivalence class. They all "point at" the same place, and the real number is the bag of all such sequences.

The same \sqrt{2}, pinned two ways

Why the two constructions feel different

Dedekind is static; Cauchy is dynamic. A Dedekind cut is a finished object: one pair of sets, no time component, no "approaching." You hand me a partition of \mathbb{Q}, I hand you a real number. Order is built in — \alpha < \beta means L_\alpha \subsetneq L_\beta — so inequalities are proper subset relations, and the supremum of a bounded family is just the union of the L-halves. The whole theory is set-theoretic.

A Cauchy sequence is a process: an infinite list, unfolding in time, with terms that agree to more and more digits. Equivalence is the business of "two processes aimed at the same target." Addition is pointwise: (a_n) + (b_n) = (a_n + b_n). Multiplication is pointwise too. Limits are just limits of sequences — you stay inside the language of sequences for every construction. The whole theory is analytic.

Dedekind wants order; Cauchy wants distance. Dedekind's machinery needs only the relation < on \mathbb{Q}. It does not care about "how far apart" two rationals are, only about which one is bigger. That is why Dedekind cuts are the cleanest proof of the supremum property: supremum is an order concept, and Dedekind's reals are just a cleaned-up order on pairs of subsets.

Cantor's machinery needs a metric — a notion of distance. A Cauchy sequence is one where the distance between late terms shrinks. The equivalence class lives inside the metric. This means the Cauchy construction generalises: swap \mathbb{Q} with any metric space that has holes in it (any incomplete metric space), and the same recipe produces its completion. The integers completed this way give nothing new. The rationals with the usual distance give \mathbb{R}. The rationals with a p-adic distance give the p-adic numbers \mathbb{Q}_p — a different completion, a different field, used in number theory. Dedekind's construction cannot do this; it is specific to ordered fields.

Dedekind is easier for beginners; Cauchy is easier for calculus. If you want to prove the supremum property cleanly, use Dedekind: the supremum is literally the union of the lower halves, and no \varepsilon-N argument appears. If you want to use real numbers in analysis — to prove the Bolzano–Weierstrass theorem, define \int_a^b f(x)\,dx as a limit of Riemann sums, or show that e^x is continuous — use Cauchy: limits, continuity and convergence are already written in the native language of sequences.

Why they produce the same ℝ

Here is the deep fact. An ordered field is a set with addition, multiplication and an order, obeying the usual rules (the field axioms plus order compatibility). It is complete if every non-empty subset bounded above has a supremum, or equivalently (over an Archimedean field) if every Cauchy sequence converges.

The proof sketch: you first identify \mathbb{Q} inside both F and F' (every ordered field contains a copy of \mathbb{Q}). Then, for any \alpha \in F, the set \{q \in \mathbb{Q} : q < \alpha\} is bounded above in F' — so it has a supremum there. Define \varphi(\alpha) to be that supremum. Check that \varphi is a ring homomorphism, order-preserving, and a bijection — all follow from completeness on both sides.

The Dedekind construction produces a complete ordered field. The Cauchy construction produces a complete ordered field. By the uniqueness theorem, they must be isomorphic. So "the reals built by Dedekind" and "the reals built by Cantor" are the same field, just written in different outfits.

Why completeness forces this uniqueness: completeness fixes where every bounded set has to "end." Once you have the rationals sitting inside and you have decided that every bounded set reaches its supremum, there is no room for two different numbers to play the same role — the supremum pins down a unique element.

When each picture is more natural

Use Dedekind when the problem is about order.

• Proving the supremum property directly. The supremum is the union of L-halves. Two lines of set theory.
• Showing \sqrt{2} exists. The cut L = \{q : q^2 < 2 \text{ or } q < 0\} is visibly well-defined; you can see the gap.
• Building surreal numbers and other order-based extensions. Dedekind-style partitions generalise to totally ordered sets that are not fields.

Use Cauchy when the problem is about limits or distance.

• Completing a metric space in general. Every metric space has a Cauchy completion. The rationals happen to complete to \mathbb{R}; other metrics give \mathbb{Q}_p, \ell^2, C[0,1], and so on. This is the bedrock of functional analysis.
• Writing down a specific real number algorithmically. A computer stores \sqrt{2} as a sequence — 1.41421356\ldots truncated to 64 bits. That is a Cauchy representative, not a Dedekind cut.
• Proving anything that involves convergence. The intermediate value theorem, the extreme value theorem, Taylor's theorem — the proofs use sequences, so Cauchy is the native language.

A worked check that the two \sqrt{2}'s agree

Take the Dedekind cut L = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\} and the Cauchy sequence s = (1, 1.4, 1.41, 1.414, 1.4142, \ldots). You want to show they represent the same real number. Under the uniqueness theorem, the map is: given the Cauchy class [s], compute

— the set of rationals eventually overtaken by the sequence.

For s = 1, 1.4, 1.41, 1.414, \ldots: a rational q is eventually below some s_n iff q < \sqrt{2}, iff q < 0 or q^2 < 2. That is exactly L. The two portraits of \sqrt{2} coincide, pointwise, as subsets of \mathbb{Q}. Same number, two descriptions.

What to take away

There is only one \mathbb{R}, but there are two (in fact several) ways to build it. Dedekind's construction treats a real as a boundary in the rationals — a way of slicing \mathbb{Q} into below-and-above. Cantor's treats a real as a destination — an equivalence class of rational sequences that are all heading the same way. Both are correct; both produce the unique complete ordered field; and the choice between them is a matter of which side of the coin is more useful today.

The payoff for understanding both is that you can read whichever construction the next textbook you open happens to use, and you can pick the one that makes your current problem simpler. Proving a supremum exists? Reach for Dedekind. Proving a function is continuous? Reach for Cauchy. Under the hood, they are the same real numbers — just described by two different but equally legitimate nineteenth-century geniuses.

Related: Real Numbers — Properties · Dedekind Cut Visualiser · Is √2 a Real Number or Just a Symbol? · Completeness Axiom Explained