$\displaystyle [a_n]$ and $\displaystyle [b_n]$ be two sequences

Let Limit n->oo ($\displaystyle a_n-b_n$) = 0 (i.e Limit n tends to infinity, an=bn)

Is Limit n->oo ($\displaystyle a_n^2 -b_n^2)$ =0?

(I'm trying to prove that there might exist nested rational interval with no rational point - I need the above for that. Essentially I have setup $\displaystyle a_n^2<2$ and $\displaystyle b_n^2>2$, thus will show that if x belongs to this interval then $\displaystyle x^2=2$ => $\displaystyle x$ is not rational)

Thanks