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Cauchy sequence
This term is a technical designation used exclusively within mathematical analysis and topology. It describes a sequence where the elements become arbitrarily close to each other as the sequence progresses, regardless of whether they converge to a specific limit within the given space.
In the context of complete metric spaces, every Cauchy sequence is guaranteed to converge. This property is central to the definition of completeness, making the term indispensable when discussing the real numbers or Banach spaces. It is never used in casual conversation or non-mathematical contexts.
Meanings
A sequence of elements in a metric space such that for every positive real number epsilon, there exists a natural number N such that the distance between any two elements in the sequence beyond the Nth term is less than epsilon.
The convergence of a Cauchy sequence is a fundamental property of complete metric spaces.
Examples
A cauchy sequence in a complete metric space always converges.
Do you understand how to prove that this is a cauchy sequence?
I wonder if every cauchy sequence of rational numbers converges to a real number.
The professor explained the relationship between a convergent sequence and a cauchy sequence.
The professor explained the relationship between a convergent sequence and a cauchy sequence.
We can define the completion of a metric space using equivalence classes of cauchy sequences.
We can define the completion of a metric space using equivalence classes of cauchy sequences.
Wait, is every convergent sequence necessarily a cauchy sequence?
Wait, is every convergent sequence necessarily a cauchy sequence?
The proof relies on the fact that the given series forms a cauchy sequence.
Let us assume that the sequence is a cauchy sequence and see where that leads.