Let $\{u_n\}$ and $\{v_n\}$ be two sequences satisfying the conditions
$\displaystyle\lim_{n\rightarrow\infty}|u_n-v_n|=0$,
$\displaystyle\lim_{n\rightarrow\infty}|u_n|=\lim_{n\rightarrow\infty}|v_n|=+\infty$.
Prove that $ \lim_{n\rightarrow\infty}\left(\frac{u_n}{|u_n|}-\frac{v_n}{|v_n|}\right)=0. $