TITLE: Space with non-convergent Cauchy sequence
QUESTION [7 upvotes]: 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?

REPLY [10 votes]: 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} $.