Understanding the Cauchy Sequence Definition in a Metric Space

The question describes a property of a sequence \( (x_n) \) in a metric space \( (E, d) \). A metric space is a set \( E \) where a distance function \( d \) is defined between any two points, satisfying certain properties (non-negativity, identity of indiscernibles, symmetry, and triangle inequality).

The given condition is: for each \( \epsilon > 0 \), there exists a natural number \( n_\epsilon \) such that for all natural numbers \( m \) and \( n \) greater than or equal to \( n_\epsilon \), the distance between \( x_m \) and \( x_n \) is less than \( \epsilon \). Mathematically, this is written as:

\( \forall \epsilon > 0, \exists n_\epsilon \in \mathbb{N} \text{ such that } m, n \in \mathbb{N}, m \ge n_\epsilon, n \ge n_\epsilon \implies d(x_m, x_n) < \epsilon \)

Identifying the Sequence Type: Cauchy Sequence

This specific condition is the fundamental definition of a Cauchy sequence in a metric space. A Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. This is different from a convergent sequence, where elements get arbitrarily close to a specific limit point.

• A sequence \( (x_n) \) is convergent if there exists a point \( x \in E \) such that for every \( \epsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( n \ge N \), \( d(x_n, x) < \epsilon \).
• A sequence \( (x_n) \) is a Cauchy sequence if for every \( \epsilon > 0 \), there exists \( n_\epsilon \in \mathbb{N} \) such that for all \( m, n \ge n_\epsilon \), \( d(x_m, x_n) < \epsilon \).

Cauchy Sequences and Completeness

In any metric space, every convergent sequence is a Cauchy sequence. However, the converse is not always true. A metric space where every Cauchy sequence converges to a point within that space is called a complete metric space. For example, the set of real numbers \( \mathbb{R} \) with the standard distance \( d(x, y) = |x-y| \) is a complete metric space, so in \( \mathbb{R} \), Cauchy sequences and convergent sequences are the same. But in the set of rational numbers \( \mathbb{Q} \), the sequence defined by \( x_n = (1 + 1/n)^n \) is a Cauchy sequence but does not converge to a rational number (it converges to \( e \), which is irrational). Therefore, \( \mathbb{Q} \) is not a complete metric space.

The condition provided in the question precisely matches the definition of a Cauchy sequence.

Conclusion

The sequence of points in the metric space satisfying the given condition is by definition a Cauchy sequence.