Cauchysorozat

Cauchysorozat, or Cauchy-sorozat, is a term used in mathematics to describe a sequence whose terms become arbitrarily close to each other as the index grows. Formally, a sequence (x_n) in a metric space (X,d) is a Cauchy sequence if for every ε > 0 there exists N such that for all m,n ≥ N we have d(x_m, x_n) < ε.

In the real numbers with the usual metric, being a Cauchy sequence is equivalent to the sequence being convergent. That is, every convergent sequence is Cauchy, and every Cauchy sequence converges to a real limit. This property reflects the completeness of the real numbers. By contrast, the rational numbers are not complete: there exist Cauchy sequences of rational numbers that do not converge to a rational limit. A classical example is a rational sequence that approximates sqrt(2); it is Cauchy in Q but converges to an irrational number outside Q.

The concept generalizes to normed and metric spaces: a sequence is Cauchy if the distance between its terms tends to zero as the indices increase. A metric (or normed) space is called complete if every Cauchy sequence in it converges within the space. The real numbers are complete; many function spaces and Banach spaces are complete as well, which underpins much of analysis and functional analysis.

Cauchy sequences provide a foundational tool for defining limits, continuity, and convergence in spaces where direct limit arguments may be difficult. They also underpin the construction of the real numbers via completion of the rationals.