Suppose that there exists $\displaystyle N \in N$, such that $\displaystyle |a_n-a_m|<1 $for all $\displaystyle n,m>=N$ (this will be true, in particular, if $\displaystyle (a_n)$ is cauchy). Prove that $\displaystyle (limsup_{n-->\infty}a_n)-(liminf_{n-->\infty}a_n)$$\displaystyle <=2$.

it seems like an easy proof, but i can't find the connection between limsup and cauchy sequence, need some help.