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?