My question concerns the interpretation of the nested interval property in Understanding Analysis by Abbott (2015).
I would appreciate some input from members of the community here as to whether my understanding of it is appropriate. That is because it forms part of a proof that I am struggling to understand, and which I will ask in a separate question.
Context.
The statement of the nested interval property I am using from Abbott is:
Theorem 1.4.1. (Nested Interval Property). For each $n \in \mathbb{N}$, assume we are given a closed interval $I_n = [a_n, b_n] = \{x \in \mathbb{R} : a_n \leq x \leq b_n \}$. Assume also that each $I_n$ contains $I_{n+1}$. Then the resulting nested sequence of closed intervals
$$I_1 \supseteq I_2 \supseteq I_3 \supseteq I_4 \supseteq \cdots$$
has a nonempty intersection; that is, $\bigcap^{\infty}_{\space n=1} I_n \neq \emptyset$.
The proof uses the axiom of completeness to show that there exists an $x \in \bigcap^{\infty}_{n=1} I_n$, where $x \in \mathbb{R}$.
Query.
Does this theorem admit the interpretation that any real number $x \in \mathbb{R}$, as an entity in a continuum, can be characterised by the property that it belongs to the intersection of a nested infinite sequence of closed non-empty bounded intervals of real numbers?
I am not experiencing difficulty with understanding the proof or the theorem, in the sense that it is a statement about the intersection of a nested infinite sequence of closed non-empty bounded intervals of real numbers being non-empty.
And to be clear, I am not asking whether the interpretation that I have outlined is useful for characterising the reals $\mathbb{R}$ in any way, as I have heard that modern definitions do this using Dedekind cuts, rather, whether it is a faulty intepretation. And if the question seems trivial, it might be helpful to bear in mind that I have a pre-analysis (i.e. high school/16th century possibly earlier) understanding of real numbers, where real numbers are these point-like entities whose existence I take for granted.
Best Answer
The notion of bounded intervals $I_n\,=\,\{x: a_n \leq x \leq b_n, -\infty <a_n \leq b_n < \infty \}$, each one a subset of the last, and such that $d_n\,=\,b_n-a_n\,\to\,0,$ can be the basis of a characterisation of real numbers. Here I follow the development in Knopp's book Theory and Application of Infinite Series.
Define a nest of intervals to be a sequence of $I_n=[a_n,b_n]$ with rational endpoints, always $I_{n+1}\subset I_n,$ and $d_n \to 0$ as $n \to \infty.$
Then the following can be shown.
There is at most one rational $r$ belonging to every $I_n$ of a given nest, and if there is one, we can say $r$ is defined by the nest, and we may view the nest $(I_n)$ as "another symbol for the number $r$."
There exist nests with no rational $r$ as in the previous item.
Since a nest with no $r$ is in a sense unoccupied, we can try to give some meaning to the idea that this given nest is a number.
As in the case of Dedekind cuts, there is much to check to make certain that this outline yields a complete ordered field, containing and extending the rational field. This verification is shown in some detail within the first chapter of Knopp's book.
Let's again go to your query. You specifically asked about intervals with real end-points. By completeness, a nest of real-endpoint intervals (again with $d_n \to 0$ as $n \to \infty$) strikes a unique real number. The characterization you proposed is a bit trivial, because any interval is already a subset of the real numbers, and we have already taken the real numbers for granted. But it is true, if we add the shrinking-to-nothing mentioned by Renfro in the comments. If $\xi$ is real, there exists such a nest. If a nest is given, it contains a real.
Your theorem 1.4.1 is not the place to seek a characterization of an arbitrary real. But this theorem shares the idea of nested intervals with a valid construction of the real number system.