# limsup and liminf of sequence of intervals in $\mathbb R$ (2)

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?

@Didier: For example how can I find $\cap_{n=k}^{\infty} I_n$ and $\cup_{n=k}^{\infty} I_n$ ? –  passenger May 12 '12 at 10:47
Surely you spotted that $I_k\approx(2,10)$ or $[2,10]$ or $(2,10]$ or $[2,10)$ when $k\to\infty$? Basically, the task is to determine which one is correct. –  Did May 12 '12 at 11:43
What does $\bigcap_{n=0}^{n=100} \bigcup_{n=0}^{n=100} I_n$ mean? You can't use the same index variable $n$ in both operators. –  Greg Martin May 13 '12 at 4:52