In Prof. Folland textbook Adavanced Calculus
$\textbf{1.20 Theorem}$. A sequence $\{x_k\}$ in $\mathbb{R^n}$ is convergent if and only if it is Cauchy.
In this theorem, if discussing $\mathbb{R}^n$, since it is complete, so convergent is Cauchy and vice versa. However, usually we cannot say a sequence being Cauchy is convergent.
For example, $R(0,1]$ is not a complete space?
When I read this book, I am confused about this.
Ex:$\{\frac{1}{k}\},\ \ k\in N^{++}$, the $\text{inf}\{\frac{1}{k}\} = 0 \notin R(0,1]$
It seems that this textbook does not emphasize this. When I read this textbook, I am confused about this.
There are many metric spaces in which Cauchy sequences do not converge. In particular, subsets of $\mathbb R$ which are not closed are not complete, so there exist Cauchy sequences in that subset which do not converge.
I like to think that nonconvergent Cauchy sequences are sequences which ought to converge, but cannot because there is a missing point.