Does there exist a convergent sequence $\{a_n\}$ such that $n d(a_{n+1},a_n)$ is not bounded? Does there exist a convergent sequence $\{a_n\}$ such that "$n d(a_{n+1} , a_n)$ is not bounded?
If not, how can i prove that $n d(a_{n+1} , a_n)$ is bounded? And is it right to conclude this;
This happens because domain of sequence is $\mathbb{N}$?

For example, let $a_{n}=(-1)^{n}\frac{1}{\sqrt{n}}$. Then $\lim_{n\rightarrow\infty}a_{n}=0$ and $nd(a_{n+1}-a_{n})=n(\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}})$ come to infinity as $n$ come to infinity.
