Are there Cauchy sequence that are not bounded?

Question

I was wondering, Is there Cauchy sequence that are not bounded ? Of course, in complete spaces is not possible. I have a theorem that says that if $(A,d)$ is not complete, then $(\bar A,d)$ is complete. But are there spaces s.t. indeed for all $\varepsilon>0$, there is $N$ s.t. $d(x_n,x_{m})<\varepsilon$ for all $n\geq N$, and all $r\geq 0$, but the ball also move to $+\infty $ ?

Answer 1

No, that cannot happen. Suppose that $(x_n)_{n \geq 0}$ is a Cauchy sequence in some metric space. By definition, there exists an integer $N \geq 0$ such that for all $n,m \geq N$ we have $d(x_n,x_m) \leq 1$. Thus, all the points $\{x_n ~|~ n \geq N\}$ are contained in the ball of radius 1 around $x_N$. By adding the finitely many points $x_1,\dots,x_{N-1}$, this sequence cannot become unbounded.

Answer 2

Note that $d(x_n,x_N)<\epsilon$ for all $n\ge N$ and that only finitely many $n<N$ are to be considered beyond that.