Suppose we have a metric space $(X,d)$, can $x_n \to x' $ as $n\to \infty$ such that $x'\notin X$ for any Cauchy sequence if the interior of $X$ is nonempty?
My logic: if $X$ has nonempty interior, then we can define a constant sequence $x_n = x$ with $x\in X$. And this constant sequence is Cauchy obviously. Hence, there is no such metric space.
Is my logic correct?