cauchy sequence: meaning, definition, pronunciation and examples
C2/Extremely Rare (Technical)Formal, Academic, Technical
Quick answer
What does “cauchy sequence” mean?
A sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses, without necessarily having a known limit.
Audio
Pronunciation
Definition
Meaning and Definition
A sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses, without necessarily having a known limit.
In mathematical analysis, a sequence where for any chosen small positive distance, there exists a point in the sequence beyond which all terms are within that distance of each other. It is a fundamental concept for defining completeness in metric spaces and is essential for understanding convergence, real numbers, and calculus.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Pronunciation may have slight regional variations. The spelling and concept are identical in all English-speaking academic communities.
Connotations
Purely technical, carries the same academic weight and precision in both varieties.
Frequency
Used exclusively in university-level mathematics education and research with identical frequency.
Grammar
How to Use “cauchy sequence” in a Sentence
X is a Cauchy sequenceThe sequence (x_n) is CauchyTo prove that Y is a Cauchy sequenceSatisfy the Cauchy criterionVocabulary
Collocations
Examples
Examples of “cauchy sequence” in a Sentence
verb
British English
- The proof shows the series cauchifies (rare technical verb) in that space.
- We need to demonstrate the sequence Cauchy-converges.
American English
- The algorithm is designed to cauchyfy (rare) the generated points.
- To solve, we first prove the iterates form a Cauchy sequence.
adverb
British English
- (No standard adverbial form. Pseudo-example:) The terms approached each other Cauchily (non-standard, illustrative).
American English
- (No standard adverbial form. Pseudo-example:) The algorithm constructs points Cauchily (non-standard, illustrative).
adjective
British English
- The sequence has the Cauchy property.
- We apply the Cauchy criterion for convergence.
American English
- This is a classic Cauchy-sequence argument.
- A space where every Cauchy sequence converges is called complete.
Usage
Meaning in Context
Business
Not used.
Academic
Central concept in real analysis, functional analysis, and metric space theory. Used in lectures, textbooks, and proofs.
Everyday
Never used in everyday conversation.
Technical
Core term in mathematical research, physics, and engineering disciplines involving advanced calculus.
Vocabulary
Synonyms of “cauchy sequence”
Strong
Weak
Vocabulary
Antonyms of “cauchy sequence”
Watch out
Common Mistakes When Using “cauchy sequence”
- Pronouncing 'Cauchy' as /ˈkɔːtʃi/ or /ˈkaʊtʃi/. Correct is /ˈkəʊʃi/ or /ˈkoʊʃi/.
- Using it as a general adjective (e.g., 'The data was Cauchy') – it only modifies 'sequence' or related nouns like 'criterion', 'net'.
- Forgetting it is a proper noun and not capitalising 'Cauchy' in formal writing.
FAQ
Frequently Asked Questions
Not exactly. In a complete space (like the real numbers), they are equivalent. However, in an incomplete space, a sequence can be Cauchy (terms get arbitrarily close) but not converge because the limit point is missing from the space.
Augustin-Louis Cauchy was a pioneering 19th-century French mathematician who rigorously formulated many concepts in calculus and analysis, including this precise definition of a sequence's internal closeness, which was crucial for putting calculus on a firm logical foundation.
Yes. The sequence 1, 1/2, 1/3, 1/4, ... is Cauchy. The difference between terms 1/n and 1/m becomes smaller than any positive number you choose, provided n and m are taken large enough.
It is absolutely fundamental in Pure Mathematics (Real Analysis, Functional Analysis, Topology). It is also important in Applied Mathematics, Theoretical Physics (especially in areas dealing with spacetime and quantum mechanics), and some branches of Engineering that use advanced mathematical modelling.
Cauchy sequence is usually formal, academic, technical in register.
Cauchy sequence: in British English it is pronounced /ˈkəʊʃi ˌsiːkwəns/, and in American English it is pronounced /ˈkoʊʃi ˌsiːkwəns/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “(None – purely technical term)”
Learning
Memory Aids
Mnemonic
Think of a group of hikers (the sequence terms) walking through fog towards an unknown destination. They don't know where they're going, but they make sure to stay close to each other (the Cauchy property). If they can all get arbitrarily close, they form a Cauchy sequence.
Conceptual Metaphor
MATHEMATICAL CONVERGENCE IS A JOURNEY OF INCREASING CLOSENESS (without a specified endpoint).
Practice
Quiz
What is the primary importance of a Cauchy sequence in real analysis?