Cauchy sequence
Definition
In the field of mathematical analysis, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Named after the French mathematician Augustin-Louis Cauchy, it is a fundamental concept in the study of metric spaces and topological spaces.
Formal Definition
Given a metric space (M, d), a sequence (x_n) in M is a Cauchy sequence if for every positive real number ε > 0, there exists a positive integer N such that for all natural numbers m, n > N, the distance d(x_m, x_n) < ε.
Properties
A Cauchy sequence has several important properties that make it a useful tool in analysis. These properties include:
- Boundedness: Every Cauchy sequence is bounded. This means that there exists a real number M such that the absolute value of each term of the sequence is less than or equal to M.
- Subsequence Convergence: Every subsequence of a Cauchy sequence is also a Cauchy sequence. Furthermore, if a subsequence of a Cauchy sequence converges, then the whole sequence converges to the same limit.
- Limit Point: Every Cauchy sequence has at least one limit point in its closure.
Cauchy Sequences in Different Spaces
The behavior of Cauchy sequences can vary depending on the space in which they are considered. For example, in a complete metric space, every Cauchy sequence converges to a limit within the space. This is not necessarily the case in an incomplete metric space, where a Cauchy sequence may not have a limit within the space.
Applications
Cauchy sequences are used in various branches of mathematics, including real analysis, complex analysis, and functional analysis. They are particularly useful in proving the existence of limits and in constructing the real numbers from the rational numbers.