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Canonical static concept record

Cauchy sequences of rationals

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Summary

Rational Cauchy sequences, with two sequences identified when their difference converges to zero.

Record metadata

Canonical name: Cauchy sequences of rationals
Concept ID: rational_cauchy_sequences
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Topology, Set theory, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Quotient presentation

Carrier(s)

set of Cauchy sequences in $\mathbb Q$

Data

equivalence relation $(a_n)\sim(b_n)$ iff $a_n-b_n\to0$

Canonically induces

metric completion of $\mathbb Q$

Notes

The definition of convergence here uses the rational absolute-value metric, not a prior construction of the real numbers.

Concept sources

Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Rational numbers $\mathbb Q$ Cauchy sequences of rationals

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: rationals_to_cauchy_sequences
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form Cauchy sequences

Authored explanation

Take rational Cauchy sequences in the absolute-value metric.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals
nLab — real number — real number · mathematical reference · source ID nlab-real-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.