I am trying to understand how important is the distinction between Cauchy sequences and convergent sequences in normed vector spaces $E$. So far I have only come across examples where the Cauchy sequence $\{x_n\}$ where $x_n\in E$ fails to converge only because the limit point is not in $E$ and an extension to $E$ typically by completion fixes the problem. For example:
$$x_n\colon[0,1]\to\Bbb R, \quad t\mapsto \sum_{k=0}^n\frac{t^k}{k!},$$
where $E\triangleq \mathcal{P}([0,1])$ is the space of polynomial functions on $[0,1]$ with uniform convergence norm. I want to know if this is the only kind of failure mode for the convergence of a Cauchy sequence.
Best Answer
This can be viewed as "the only kind of failure mode", since any non-complete metric space can be viewed as a subspace of a complete metric space. For more details, see e.g. this question and its answers.