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Is it true that $x_n$ is a Cauchy sequence in some Hilbert space if and only if:

for all $m \in \mathbb{N}$, for all $\epsilon > 0$, there exists $N=N(m) \in \mathbb{N}$ (so the $N$ can depend on $m$) such that $$\lVert x_{n+m}-x_n \rVert \leq \epsilon \quad\text{ if $n \geq N(m)$}.$$

Is that correct?

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  • @Gae.S. I think so and it seems what I asked for is not true. I read this answer https://math.stackexchange.com/a/1836672/302620 which says it is true so I wanted to check – StopUsingFacebook Feb 28 '20 at 15:58
  • Thanks. ....... – StopUsingFacebook Feb 28 '20 at 16:55
  • User ClementC. is correct in his remark about the answer you have linked. The author of the answer said a couple of times that the condition does not imply convergence. However, he clearly misread the question, therefore his intentions and perhaps the value he gives to convergence vs Cauchy are unclear. –  Feb 28 '20 at 18:16

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