Example of collection of nested intervals whose intersection is empty

Question

Give an example of intervals not empty $I_n \in \mathbb{R} (n ∈ \mathbb{N})$ which are bounded, with $I_{n + 1} \subset I_n$ for each $n$, and $\cap_{n = 1}^\infty I_n=\emptyset$.

I thought about $[n,\infty)$, but this is not bounded.

Answer 1

For example $I_n=(0,\frac1n)$.

It is probably an example near some theorems about compactness, showing that the assumption, that the set are closed, is important.

Answer 2

If the intervals were closed, the result will not be true (it's called Cantor's theorem, see here). Basically, it follows from the compacity.

So you have to choose open intervals. An easy choice can be the following:

$I_n = (0,1/n)$

The intersection of all $I_n$ is empty, because if there were $x \in \cap I_n$, $x$ will be strictly greater than zero, and there will exist $n$ such that $x>1/n$, because $1/n$ converges to zero, so for this $n$, $x \notin I_n$.