A **Cauchy sequence** is a sequence in a metric space (or more generally in a topological space) where the elements of the sequence become arbitrarily close to each other as the sequence progresses. In other words, a sequence \((a_n)\) is called a Cauchy sequence if, for every \(\varepsilon > 0\), there exists a positive integer \(N\) such that for all \(m, n \geq N\), the distance between \(a_m\) and \(a_n\) is less than \(\varepsilon\).

### Formal Definition

Let \((a_n)\) be a sequence in a metric space \((X, d)\). The sequence \((a_n)\) is called a **Cauchy sequence** if:

\[
\forall \varepsilon > 0, \, \exists N \in \mathbb{N} \, \text{such that} \, \forall m, n \geq N, \, d(a_m, a_n) < \varepsilon.
\]

Here:

- \(d(a_m, a_n)\) denotes the distance between \(a_m\) and \(a_n\) in the metric space \(X\).
- \(\varepsilon > 0\) is an arbitrarily small positive number.
- \(N\) is a positive integer after which all terms \(a_n\) are within a distance \(\varepsilon\) of each other.

### Key Points

1. **Convergence**: In a **complete** metric space (like the real numbers \(\mathbb{R}\)), every Cauchy sequence is **convergent**; that is, it has a limit within the space. However, not every Cauchy sequence converges in spaces that are not complete (e.g., the rational numbers \(\mathbb{Q}\)).

2. **Intuition**: The idea is that the terms of the sequence get closer and closer together as the sequence progresses. However, unlike convergence, we do not specify any particular point they get close to; we only require that the terms get close to each other.

3. **Importance in Analysis**: Cauchy sequences are critical in real analysis and functional analysis because they provide a way to define the concept of completeness of a space.

### Example

Consider the sequence \((a_n)\) defined by \(a_n = \frac{1}{n}\) in the real numbers \(\mathbb{R}\):

- For any \(\varepsilon > 0\), choose \(N > \frac{1}{\varepsilon}\). Then, for all \(m, n \geq N\),
  
  \[
  |a_m - a_n| = \left| \frac{1}{m} - \frac{1}{n} \right| \leq \frac{1}{N} < \varepsilon.
  \]

- Hence, \((a_n)\) is a Cauchy sequence.

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### Related Questions

1. What is the difference between a Cauchy sequence and a convergent sequence?
2. What is an example of a Cauchy sequence that is not convergent?
3. What is a complete metric space, and why is completeness important for Cauchy sequences?
4. How do Cauchy sequences relate to the concept of uniform convergence?
5. Can a Cauchy sequence exist in any metric space, and what conditions affect its convergence?

### Tip

Understanding Cauchy sequences can provide insight into the foundational properties of metric spaces, including convergence and completeness, which are essential concepts in analysis.

Learn what a Cauchy sequence is in the context of metric spaces and real analysis. Understand the formal definition, key theorems like the Cauchy Sequence Theorem, and examples demonstrating the concept of convergence and completeness. Suitable for undergraduate mathematics students.

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