Let $a_n$ a sequence with $a_n = \sin(\frac{n\pi}{4}+\frac{\pi}{4})$ how exactly do I calculate the liminf and limsup of this one?
Thoughts: I obviously went down the typical way took $a_{2n}=\sin(\frac{n\pi}{2}+\frac{\pi}{4})$ and $a_{2n+1}=\sin(\frac{n\pi}{2}+\frac{3\pi}{4})$ but im actually stuggling to calculate these trigonometrical things could i have a help solving these?
ps: I can solve the rest my self I am just stuck in the trigonometries somehow.
That sequence contains five and only five numbers: $0$, $\pm1$, and $\pm\frac1{\sqrt2}$. Each of them appears infinitely often. So$$\liminf\nolimits_na_n=-1\quad\text{and}\quad\limsup\nolimits_na_n=1.$$