Cauchy Sequence

A sequence $x_n$ in a metric space $(X,d)$ is called a Cauchy sequence if for every $\epsilon >0$ there exists an integer $N$ such that for all $m,n\ge N$, the distance between the terms $x_m$ and $x_n$ is less than $\epsilon$; that is,

$$\forall \epsilon >0\exists N\in \mathbb{N}\text{s.t.}m,n\ge N⟹d(x_m,x_n)<\epsilon .$$

Intuitively, this means that as the sequence progresses, its elements get arbitrarily close to one another, regardless of what value they approach or even whether they appear to approach a limit within the space.

The concept of a Cauchy sequence is fundamental because it captures the idea of convergence without requiring explicit knowledge of a limit point. A metric space in which every Cauchy sequence converges to a point in that space is called complete. For example, the real numbers (\mathbb{R}) are complete, so every Cauchy sequence of real numbers converges to a real number. In contrast, the rational numbers (\mathbb{Q}) are not complete: there exist Cauchy sequences of rationals that converge to irrational limits, demonstrating that the limit need not lie in the original space.