$U$ is open then find a sequence of closed sets $ I_{n+1} \subset I_{n}$

Question

Let $U = ]0,1[$, then I am trying to find a sequence of closed sets $(I_n)_{n \in \mathbb{N}}$ of $\mathbb{R}$ such that :

$$ I_{n+1} \subset I_n \text{ and } \cap_{n \in \mathbb{N}} I_n \subset U$$

And such that there isn’t an $ r \in \mathbb{N}$ such that :

$$\forall N > r, I_N \subset U$$

I didn’t succeed in finding such a sequence of closed sets.

The main problem is that the intersection of closed sets is still a closed set hence the second condition makes it hard.

So for exemple : $I_n =[1/2, 1+1/n]$ don’t work since $I_{\infty} = [1/2,1]$ wich is closed and hence not a subset of $U$.

Answer 1

Such a sequence exists.

First note that since $\mathbb{R} \backslash U$ is not compact then by Borel- Lebesgue it is possible to find a sequence that fulfills the condition of the problem.

Simply take :

$$I_n = [n, +\infty)$$

Then we clearly have :

$$\bigcap_{i = 1}^{\infty} I_i = \emptyset \subset U$$

Hence such a sequence exists.