I am wondering what are the relationships of the following three statements (the sequence $\{ a_{k} \} \in \mathbb{C}^{N}$ here is assumed to be bounded):
- $\{ a_{k} \}$ is a Cauchy sequence.
- $\sum_{k \in \mathbb{N}} \Vert a_{k+1} - a_{k} \Vert^{2} < \infty$.
- $\sum_{k \in \mathbb{N}} \Vert a_{k+1} - a_{k} \Vert < \infty$
Right now, I only know that 3. implies 1. Without the boundedness assumption, 2. does not imply 1. But what if there is boundedness assumption?
Even if the $a_k$ are bounded, (2) does not imply (1).
Say $$a_k=\sum_{n=1}^k\pm\frac1n.$$
Then (2) holds regardless of how we choose the $\pm$ signs. I imagine you noticed that if we take all plus signs then $(a_k)$ is not Cauchy, but also not bounded. If we take alternating plus and minus signs then $(a_k)$ is Cauchy, by the alternating series test.
So do something in between: Take enough plus signs until you get an $a_{k_1}>1$, then switch to minus signs until you get to $k_2>k_1$ with $a_{k_2}<0$, then switch back to plus signs until $a_{k_3}>1$... Then $(a_k)$ is not Cauchy, because $\limsup a_k\ge1$ and $\liminf a_k\le0$, but we have $-1\le a_k\le 2$ for all $k$.
Edit: Oh. Quicker but less interesting: Let $a_k=\sin(\log(k))$.