Cauchy Sequence

Definition: Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. It can be defined for various spaces, such as:

For real numbers, a sequence $\langle x_n\rangle$ of real numbers, such that:

$\qquad\quad\langle x_n\rangle = x_1, x_2, x_3, \cdots$ where, $x_1, x_2, x_3, \cdots\;\in\;\mathbb{R}$

is a Cauchy sequence, if for every positive real number $\epsilon$, $\exists$  an $N\;\in\;\mathbb{N}$, such that

$\qquad\quad |x_n - x_m| < \epsilon \;\;\forall\;\; n, m\;\geq\; N$

Similarly, for rational numbers, a sequence $\langle x_n\rangle$ of rational numbers, such that:

$\qquad\quad\langle x_n\rangle = x_1, x_2, x_3, \cdots$ where, $x_1, x_2, x_3, \cdots\;\in\;\mathbb{Q}$

is a Cauchy sequence, if for every positive rational number $\epsilon$, $\exists$  an $N\;\in\;\mathbb{N}$, such that

$\qquad\quad |x_n - x_m| < \epsilon \;\;\forall\;\; n, m\;\geq\; N$


For example:

$\textrm{If}\;\;\;\quad \langle x_n\rangle =\left\{\dfrac{1}{n}\;\big|\;\; n\;\in\;\mathbb{Z}\right\}\\
\begin{align}\textrm{Then,}\; |x_n - x_m| &= \left| \dfrac{1}{n} - \dfrac{1}{m}\right|\\[6pt]
& < \dfrac{1}{n} \text{&} \dfrac{1}{m}\end{align}$

Choosing:
$\begin{align}\qquad\quad & N > \dfrac{1}{\epsilon}\\
\Rightarrow\quad\;\; &\dfrac{1}{N} < \epsilon\\
\therefore\quad\;\;\; &|x_n - x_m|\;\;< \; \dfrac{1}{n} \text{&} \dfrac{1}{m} < \;\epsilon\;\;\forall\;\; n, m\;\geq\; N \;\left(\textrm{for}\; N > \dfrac{1}{\epsilon}\right) \end{align}$

Hence, $\langle x_n\rangle =\left\{\dfrac{1}{n}\;\big|\;\; n\;\in\;\mathbb{Z}\right\}$ is a Cauchy sequence.