nLab Cauchy sequence

Idea

A Cauchy sequence is an infinite sequence which ought to converge in the sense that successive terms get arbitrarily close together, as they would if they were getting arbitrarily close to a limit. Among sequences, only Cauchy sequences will converge; in a sequentially complete space, all Cauchy sequence converge.

Definitions

The precise definition varies with the context.

A sequence $(x_i)_i$ of real numbers is Cauchy if, for every positive number $\epsilon$, almost all terms are within $\epsilon$ of one another. Explicitly:

$$\forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; |x_i - x_j| \lt \epsilon .$$

In a metric space, a sequence $(x_i)_i$ is Cauchy under the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; d(x_i,x_j) \lt \epsilon .$$

The same definition immediately applies to an extended quasipseudometric space (aka a Lawvere metric space), or anything in between.

In a gauge space, a sequence $(x_i)_i$ is Cauchy if this condition is satisfied for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; d(x_i,x_j) \lt \epsilon .$$

In a rational or real premetric space, a sequence $(x_i)_i$ is Cauchy if this condition is satisfied for the premetric for all positive rational numbers $\epsilon$. Explicitly:

$$\forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; x_i \sim_\epsilon x_j .$$

In a uniform space or preuniform space, a sequence $(x_i)_i$ is Cauchy if an analogous condition is satisfied for each entourage $U$. Explicitly:

$$\forall U,\; \exists N,\; \forall i, j \geq N,; x_i \approx_U x_j .$$

More generally, let $S$ and $T$ be sets and let $R(a, b, t)$ be a ternary relation indexed by $a, b \in S$ and $t \in T$. A sequence $(x_i)_i$ in $S$ is Cauchy if this condition is satisfied for each element $t \in T$. Explicitly:

$$\forall t,\; \exists N,\; \forall i, j \geq N,\; R(x_i, x_j, t) .$$

In a Cauchy space, a sequence $(x_i)_i$ is Cauchy if it generates a Cauchy filter. Explicitly:

$$\{ A \;|\; \exists N,\; \forall i, j \geq N,\; x_i \in A \} \in \mathcal{C} ,$$

where $\mathcal{C}$ is the collection of Cauchy filters that defines the structure of the Cauchy space.

All of the above are in fact special cases of this.

Generalizations

Multivalued sequences

A multivalued sequence on a set $A$ is a function $f:\mathbb{N} \to \mathcal{P}(T)$ such that for every natural number $n \in \mathbb{N}$, $f(n)$ is an inhabited subset of $A$.

A multivalued sequence $x:\mathbb{N} \to \mathcal{P}(\mathbb{R})$ of real numbers is Cauchy if, for every positive number $\epsilon$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist real numbers $a, b$ such that $x(i)(a)$ and $x(j)(b)$ holds and $\vert a - b \vert \lt \epsilon$.

In a metric space $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is Cauchy if, for every positive number $\epsilon$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $d(a, b) \lt \epsilon$.

In a gauge space $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is Cauchy if, for every positive number $\epsilon$ and every gauge $d:S \times S \to \mathbb{R}_{\geq 0}$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $d(a, b) \lt \epsilon$.

In a rational or real premetric space $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is Cauchy if, for every positive number $\epsilon$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $a \sim_\epsilon b$.

In a uniform space or preuniform space $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is Cauchy if, for every entourage $U$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $a \approx_U b$.

More generally, let $S$ and $T$ be sets and let $R(a, b, t)$ be a ternary relation indexed by $a, b \in S$ and $t \in T$. A multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is Cauchy if, for every element $t \in T$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $R(a, b, t)$ holds.

Nets and filters

A net is a generalization of a sequence; the definitions above serve to define a Cauchy net without any change, other than allowing $(x_i)_i$ to be a net. This is precisely the structure of a Cauchy space; instead of defining Cauchy nets in terms of Cauchy filters as above, we may equally well define a Cauchy filter to be a proper filter whose canonical net? is Cauchy.

Note that in a complete space, every Cauchy net has a limit, not just every Cauchy sequence. Rather, a space in which every Cauchy sequence converges is called sequentially complete. Note that a metric space (or even a Lawvere metric space) is in fact complete if it is sequentially complete (although this result is not valid in some weak foundations); in particular, the real line $\mathbf{R}$ is complete.

When Bill Lawvere idenitified Lawvere metric spaces with enriched categories over the closed monoidal poset $(\mathbf{R}^+,+)$, he identified Cauchy sequences in such spaces with certain adjunctions of bimodules, enough so that a metric space would be a Cauchy-complete space if and only if every adjunction of bimodules is induced by an enriched functor. Generalising this condition from $(\mathbf{R}^+,+)$ to an arbitrary closed monoidal category, we have the concept of Cauchy-complete category.

Regular sequences and nets

One can also consider sequences and nets with a modulus of convergence $\alpha$ from the positive numbers or the set of entourages to the natural numbers or the directed set of a net. This is common in constructive mathematics or in other foundations which do not assume the axiom of choice.

A sequence or net $(x_i)_i$ of real numbers is regular Cauchy or modulated Cauchy if it comes with the structure of a modulus of convergence $\alpha$ for $(x_i)_i$. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; |x_i - x_j| \lt \epsilon .$$

In a metric space, a sequence or net $(x_i)_i$ is regular Cauchy or modulated Cauchy under the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; d(x_i,x_j) \lt \epsilon .$$

In a gauge space, a sequence or net $(x_i)_i$ is regular Cauchy or modulated Cauchy if this condition is satisfied for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; d(x_i,x_j) \lt \epsilon .$$

In a rational or real premetric space, a sequence or net $(x_i)_i$ is regular Cauchy or modulated Cauchy under the same condition, now relative to the premetric on that space. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; x_i \sim_\epsilon x_j .$$

In a uniform space, a sequence or net $(x_i)_i$ is regular Cauchy or modulated Cauchy if an analogous condition is satisfied for each entourage $U$. Explicitly:

$$\forall U,\; \forall i, j \geq \alpha(U),\; x_i \approx_U x_j .$$

More generally, let $S$ and $T$ be sets and let $R(a, b, t)$ be a ternary relation indexed by $a, b \in S$ and $t \in T$. A sequence $(x_i)_i$ in $S$ is regular Cauchy or modulated Cauchy if this condition is satisfied for each element $t \in T$. Explicitly:

$$\forall t,\; \forall i, j \geq \alpha(t),\; R(x_i, x_j, t) .$$

Related concepts

• sequence, net
• modulus of convergence
• metric space, gauge space, uniform space, Cauchy space
• limit of a sequence, limit of a net
• complete space
• sequentially complete space
• Cauchy approximation

References

• Wikipedia, Cauchy sequence

• Auke Booij, Analysis in Univalent Type Theory (2020) [etheses:10411, pdf, pdf]

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)