If a sequence $\{x_n\}_{n=1}^{\infty}$ of real numbers satisfies $$lim_{n\rightarrow\infty} n\Arrowvert x_n-x_{n+1}\Arrowvert=0$$,can we prove that it is a convergent sequence?
I have tried to prove it is a Cauchy sequence but it seems hard to deal with the part of $\frac{1}{n}$.