Consider the sequence of intervals $$ { I_n = \left( 2+\frac{(-1)^n}{n+1} , 10 +\frac{(-1)^{n+1}}{n+1} \right) }$$ Find $ \limsup I_n$, $\liminf I_n$, $\bigcup_{n=0}^{n=100} \bigcap_{n=0}^{n=100} I_n$, $\bigcap_{n=0}^{n=100} \bigcup_{n=0}^{n=100} I_n$.
For $\limsup I_n , \liminf I_n$ I tried to find the unions and the intersections finding first the unions and the intersections of $ I_{2k}$ and $I_{2k+1}$
Any help?
Thank's in advance!