[Math] Give an example of a nested sequence of closed but unbounded intervals which does not have a point in its intersection.

Question

I'm finding nested intervals hard to understand and I'm really stuck on this homework question. First to make sure, a closed but unbounded interval is something like this right?

$[n,\infty)$

And if so, could we show that $I_n=[n,\infty)$ is an example with no point in its intersection? I think we could use the Archimedean Property to show that for any $x \in [1,\infty)$, there exists an $m \in \mathbb{N}$ greater than $x.$ So $x \not\in I_m$. Like a proof by contradiction?

I'd like to know if my reasoning makes sense and if this is a "concrete" counterexample.

Thanks!

[EDITED]

Answer 1

Concretely consider $I_n := [n, \infty)$. By the existence of $\lfloor x+1 \rfloor$ for all real $x$, you have that for $n>\lfloor x+1 \rfloor$, $x\notin I_n$ (wich is basically the archimedian property you mentioned), thus $$\bigcap_{n\in\mathbb N} [n,\infty) = \emptyset$$ So it is a valid example. A more general class of examples is given by this proposition:

Let $(a_n)_n$ be a sequence with $\lim\limits_{n\to\infty} a_n = \infty$ then the sequences of sets $$I^1_n := [a_n, \infty)\\I^2_n := (-\infty, -a_n]\\I^3_n := \mathbb R \setminus (-a_n, a_n)$$ all fulfill $I^j_{n+1} \subset I^j_n$ and $$\bigcap_{n\in\mathbb N} I^j_n = \emptyset$$