Find $\lim \sup$ and $ \lim \inf$ of the following sequence

Question

Find $\lim \sup$ and $\lim \inf$ of the following sequence

(i) $\frac{n}{4}-[\frac{n}{4}]$

How to solve it. I am in trouble to solve them.

Answer 1

Hint

consider limits of subsequences

$(u_{4n}), \; (u_{4n+1}), \;(u_{4n+2})$ and $(u_{4n+3})$.

for the middle one. $$u_n=\frac{n}{4}-\lfloor \frac{n}{4} \rfloor$$.

$$u_{4n}=0$$ $$u_{4n+1}=n+\frac{1}{4}-\lfloor n+\frac{1}{4}\rfloor =\frac{1}{4}$$ $$u_{4n+2}=\frac{1}{2}$$ $$u_{4n+3}=\frac{3}{4}$$

thus $\liminf=0$ and $\limsup=\frac{3}{4}$.