Comparing Dedekind Cuts and Cauchy Sequences

Both constructions aim to "complete" the rationals $\mathbb{Q}$, meaning to create a number system where every "approaching" sequence or gap is fully realized as a number.

Feature	Dedekind Cuts	Cauchy Sequences
Basic object	A downward-closed set of rationals without a greatest element	A sequence $(q_n)$ of rationals that becomes arbitrarily close together
Intuition	A "cut" describes everything less than a real number	A sequence approximates a real number
Completeness	The set of cuts fills in all gaps	The set of limits of Cauchy sequences fills in all gaps
Key property	Each cut represents a real number, possibly irrational	Each equivalence class of Cauchy sequences represents a real number
Real number	A specific subset of $\mathbb{Q}$	An equivalence class of Cauchy sequences
Least Upper Bound Property	The union of a set of cuts is a cut	Every bounded increasing sequence of Cauchy classes converges

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Constructing the Isomorphism

We define a mapping:

$\Phi: \{\text{Dedekind cuts}\} \longrightarrow \{\text{Cauchy sequences modulo equivalence}\}$

This function $\Phi$ and its inverse $\Psi$ together form a bijective (one-to-one and onto) correspondence between Dedekind cuts and equivalence classes of Cauchy sequences.

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1. From Dedekind Cuts to Cauchy Sequences

Let $A \subseteq \mathbb{Q}$ be a Dedekind cut. Since $A$ is nonempty, downward closed, and has no greatest element, for each $n \in \mathbb{N}$, we can choose $q_n \in A$ such that:

$\sup A - \frac{1}{n} < q_n < \sup A$

Such a sequence $(q_n)$ is increasing and bounded above by $\sup A$, so it is a Cauchy sequence. Define:

$\Phi(A) := [(q_n)]$

Here, $[(q_n)]$ is the equivalence class of all Cauchy sequences converging to the same real number as $(q_n)$.

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2. From Cauchy Sequences to Dedekind Cuts

Let $(q_n)$ be a Cauchy sequence in $\mathbb{Q}$, with real limit $\ell \in \mathbb{R}$. Define:

$\Psi((q_n)) := \{ r \in \mathbb{Q} : r < \lim q_n \}$

This set is a Dedekind cut:

• It is nonempty because $q_n \to \ell$ and $q_n \in \mathbb{Q}$.

• It is downward closed: if $r < s < \ell$, then $r \in \Psi((q_n))$.

• It has no greatest element because $\ell \notin \mathbb{Q}$, or if it is, $\ell \notin \Psi((q_n))$.

So $\Psi((q_n)) = A$, where $A$ is the cut below the limit of the sequence.

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Proving the Isomorphism is Bijective

Injectivity of $\Phi$

Suppose $A \ne B$ are Dedekind cuts. Then there exists $r \in \mathbb{Q}$ such that $r \in A \setminus B$ or vice versa. Thus, $\sup A \ne \sup B$. So sequences $(q_n)$ from $A$ and $(p_n)$ from $B$ converge to different limits.

Hence $\Phi(A) \ne \Phi(B)$. Therefore, $\Phi$ is injective.

Injectivity of $\Psi$

Suppose $(q_n) \sim (p_n)$ are equivalent Cauchy sequences, meaning they have the same limit $\ell$. Then $\Psi((q_n)) = \Psi((p_n)) = \{ r \in \mathbb{Q} : r < \ell \}$.

Hence, $\Psi$ respects equivalence classes, and the function is well-defined.

Surjectivity of $\Phi$

Let $[(q_n)]$ be a Cauchy sequence equivalence class. Define $A := \{ r \in \mathbb{Q} : r < \lim q_n \}$. Then $\Phi(A) = [(q_n)]$, since the constructed sequence from $A$ approximates the same limit.

Thus, every real number (i.e., every class of Cauchy sequences) corresponds to a Dedekind cut. So $\Phi$ is surjective.

Surjectivity of $\Psi$

Given a Dedekind cut $A$, construct $(q_n) \in A$ such that $\sup A - 1/n < q_n < \sup A$. Then $\Psi((q_n)) = A$, so every cut arises from some sequence.

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Preservation of Arithmetic Structure

Let $[a_n]$ and $[b_n]$ be Cauchy sequences representing real numbers $a$ and $b$.

Addition

Define $c_n = a_n + b_n$. Then:

• $(c_n)$ is Cauchy.

• $\lim c_n = a + b$

• $[a_n] + [b_n] = [c_n]$, and $\Phi(A + B) = [c_n]$ matches this.

Multiplication (Assume $a, b > 0$)

Define $d_n = a_n \cdot b_n$. Then:

• $(d_n)$ is Cauchy.

• $\lim d_n = ab$

• $[a_n][b_n] = [d_n]$, and $\Phi(AB) = [d_n]$

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Summary Diagram

$$\boxed{\text{Dedekind Cut } A \xrightarrow{\Phi} (q_n) \xrightarrow{\Psi} \text{Cut } A'}$$ $$\boxed{(q_n) \xrightarrow{\Psi} A \xrightarrow{\Phi} (q_n)}$$

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Concluding Statement

Dedekind cuts and Cauchy sequences are isomorphic models of the real numbers $\mathbb{R}$. The isomorphism $\Phi$ is both injective and surjective (bijective), and it preserves arithmetic operations and order. These two views are mathematically equivalent, and both provide rigorous foundations for real analysis.