Sequence under quasi isometry

Question

Suppose ($x_n$) is a divergent sequence in $\mathbb{R}$ and $\phi$ is an Quasi-Isometry from $\mathbb{R}$ to $\mathbb{R}$ then we have to show that ($\phi(x_n)$) can not oscillate. If ($\phi(x_n)$) oscillate finitely then it will be bounded and by definition we get a contradiction but if it oscillate infinitely then i am stuck to write the rigorous argument. Intuitively it seems to be clear. Thanks for any help.