nLab
Cauchy sequence

Cauchy sequences

Idea

A Cauchy sequence is an infinite sequence which ought to converge in the sense 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(x_i)_i of real numbers is Cauchy if, for every positive number ϵ\epsilon, almost all terms are within ϵ\epsilon of one another. Explicitly:

ϵ,N,i,jn,x ix jϵ. \forall \epsilon,\; \exists N,\; \forall i, j \geq n,\; |x_i - x_j| \leq \epsilon .

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

ϵ,N,i,jn,d(x i,x j)ϵ. \forall \epsilon,\; \exists N,\; \forall i, j \geq n,\; d(x_i,x_j) \leq \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(x_i)_i is Cauchy if this condition is satisfied for each gauging distance separately. Explicitly:

d,ϵ,N,i,jn,d(x i,x j)ϵ. \forall d,\; \forall \epsilon,\; \exists N,\; \forall i, j \geq n,\; d(x_i,x_j) \leq \epsilon .

In a uniform space, a sequence (x i) i(x_i)_i is Cauchy if an analogous condition is satisfied for each entourage UU. Explicitly:

U,N,i,jn,;x i Ux j. \forall U,\; \exists N,\; \forall i, j \geq n,; x_i \approx_U x_j .

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

  • the proper filter {AN,i,jn,x iA} \{ A \;|\; \exists N,\; \forall i, j \geq n,\; x_i \in A \} is Cauchy.

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

Generalisations

A net is a generalisation of a sequence; the definitions above serve to define a Cauchy net without any change, other than allowing (x i) i(x_i)_i to be a net. This is fundamental to the concept of 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. However, a space in which every Cauchy sequence converges is called sequentially complete. Note that a metric space (or even a Lawvere metric space) is complete if and only if it is sequentially complete (although this result is not valid in some weak foundations); in particular, the real line R\mathbf{R} is complete.

When Bill Lawvere idenitified Lawvere metric spaces with enriched categories over the closed monoidal poset (R +,+)(\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 in induces by an enriched functor. Generalising this condition from (R +,+)(\mathbf{R}^+,+) to an arbitrary closed monoidal category, we have the concept of Cauchy complete category.

Revised on March 29, 2010 00:08:25 by Toby Bartels (98.19.50.118)