[Def]: Cauchy Sequence 歌西列

設 $(M,d)$ 為一 metric space, 點列 $(x_n)_{n=1}^\infty \subseteq M$

若對任何 $\varepsilon>0$ 皆存在 $N_{\varepsilon}>0$, s.t. 當 $n, m > N_{\varepsilon}$ 時 $d(x_n,x_m)<\varepsilon$

則稱 $(x_n)_{n=1}^\infty$ 為 Cauchy sequence 柯西列

[Def]: A Sequence is not a Cauchy sequence

設 $(M,d)$ 為一 metric space, 點列 $(x_n)_{n=1}^\infty \subseteq M$

$\exists\varepsilon>0$ such that $\forall n \in \N$, there exist $i,j>n$ such that $d(x_i,x_j)>\varepsilon$

• [example]: harmonic sequence is divergent