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 $\mathbb{N}$?

Hint: Let $a_n=0$ if $n$ is odd. Let $a_n$ decrease slowly to $0$  over the evens.
