Space with non-convergent Cauchy sequence

Question

Not all sequences that are Cauchy are convergent. Here is what I think the example should be. Somehow the metric space is open but does not contain its limit points. Is this the right direction of thought?

Answer 1

Just take any sequence of rational numbers that converges to an irrational number. Then the sequence is Cauchy in $ \mathbb{Q} $, but does not converge in $ \mathbb{Q} $.

Answer 2

There can be sequences that are Cauchy but do not converge; for example, the sequence $(1,\frac{1}{2},\frac{1}{3},\ldots)$ does not converge in the metric space $(0,2)$. A metric space in which every Cauchy sequence converges is said to be complete.

Answer 3

In $\mathbb{K}[X]$.

$\forall P \in \mathbb{K}[X], P = \sum\limits_{i \in\mathbb{N}} p_i X^i, \|P\|=\max\limits_{i \in\mathbb{N}}( | p_i | )$

Let $P_n = \sum\limits_{i = 1}^n \frac{X^i}{i!}$

$P_n$ is Cauchy but doesn't converge in $\mathbb{K}[X]$.