# limit superior and limit inferior

I'll show you one part. You show me another. I assume that this is $a_n=\frac{n}{3}-\left\lfloor \frac{n}{3}\right\rfloor$. Clearly, we have that $a_n\geqslant 0$ and so $\liminf\text{ }a_n\geqslant 0$ but considering that $a_{3n}=0$ we see that $\liminf\text{ }a_n=0$.