Cauchy sequences

# Cauchy sequences

## 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 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 Booij 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, a sequence $(x_i)_i$ is Cauchy if an analogous condition is satisfied for each entourage $U$. Explicitly:

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

In a preuniform space, a sequence $(x_i)_i$ is Cauchy if this condition is satisfied for each preuniformity $U$. Explicitly:

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

Generalizing both Booij premetric spaces and preuniform spaces, in a $T$-premetric space with set $T$, a sequence $(x_i)_i$ is Cauchy if this condition is satisfied for each element $U \in T$. Explicitly:

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

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

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 or sequentially Cauchy 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.