pairwise difference of a convergent sequence

Question

Suppose $\{x_j\} \subset X$ is a sequence in some normed vector space. We know is $\|x_{j+1} - x_{j}\| \le 1/j^p$ for $p > 1$, then $\{x_j\}$ is convergent. What about the other direction? That is, if we know the sequence converges to a limit $x$, what can we say about the rate of convergence of positive sequence $\{\|x_j - x_{j-1}\|\}_{j=2}^{\infty}$? It certainly converges to $0$ by triangle inequality. More specifically, is the sequence $\{ j \|x_j - x_{j-1}\|\}$ bounded?

Answer 1

There is no good converse.
For instance, $$(0,1/\sqrt{2},0,1/\sqrt{4},0,1/\sqrt{6},\ldots)$$ has a subsequence for which $n|x_n-x_{n-1}|$ behaves like $\sqrt{n}$.