Cauchy_sequence

Definition

A sequence $\left. \left( a_{n} \right) \right.$ is called a Cauchy sequence if, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that whenever $m,n \geq N$ it follows that $\left| a_{n} - a_{m} \right| < \epsilon$.

Compare this to the definition of convergence.

Theorem: Every convergent sequence is a Cauchy sequence

Proof: Assume $\left. \left( x_{n} \right) \right. \rightarrow x$. Let $\epsilon > 0$ be arbitrary. There exists an $N \in \mathbb{N}$ such that $n,m \geq N$ implies that $\left| x_{n} - x \right| < \frac{\epsilon}{2}$ and $\left| x_{m} - x \right| < \frac{\epsilon}{2}$.

By the triangle inequality

$$ \begin{aligned} \left| x_{n} - x_{m} \right| & = \left| x_{n} - x + x - x_{m} \right| \\ & \leq \left| x_{n} - x \right| + \left| x_{m} - x \right| \\ & < \frac{\epsilon}{2} + \frac{\epsilon}{2} \\ & = \epsilon \end{aligned} $$