I'm confused by limes superior, sup / supremum, same for lim inf...
First of all, is sup an abbreviation of supremum? $sup\Leftrightarrow supremum$?
I have invented a sequence myself and solved it, and I hope I did it correctly.
$$a_{n}=(-1)^{n}\left(1+\frac{1}{n}\right), n \geq 1$$
So, the supremum of this sequence is $\sup=(-1)^{2}\left(1+\frac{1}{2}\right)= 1\cdot\frac{3}{2}= \frac{3}{2}$
The infimum is $\inf=(-1)^{1}\left(1+\frac{1}{1}\right)= -2$
and now lim sup lim inf:
For an even $n$: $\limsup_{n\rightarrow\infty}\left(a_{n}\right )=\left(1+\lim_{n\rightarrow\infty}(\frac{1}{n})\right)=(1+0)=1$
For an uneven $n$: $\liminf_{n\rightarrow\infty}\left(a_{n}\right )=-\left(1+\lim_{n\rightarrow\infty}(\frac{1}{n})\right)=-(1+0)=-1$
I hope I haven't confused anything here? This is really confusing for me and I hope I did it correctly...