(Nested Interval Theorem) Let In:=[an, bn], n ∈ℕ, be closed intervals such that In ⊇In+1 for all

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(Nested Interval Theorem) Let In:=[an, bn], n ∈ℕ, be closed...

(Nested Interval Theorem) Let $I_{n}:=\left[a_{n}, b_{n}\right], n \in \mathbb{N}$, be closed intervals such that $I_{n} \supseteq I_{n+1}$ for all $n \in \mathbb{N}$ and $\left|b_{n}-a_{n}\right| \rightarrow 0 .$ Show that there is a unique $x \in \mathbb{R}$ such that $x \in I_{n}$ for all $n \in \mathbb{N}$. (Hint: Use Exercise 51 of Chapter $1 .$ )

Key Concepts

Completeness of the Real Numbers

The completeness of the real numbers is a cornerstone of analysis which ensures that every Cauchy sequence converges to a real number, and that every bounded monotone sequence has a limit. This property underpins the Nested Interval Theorem, as it guarantees that the limit of the endpoints of the nested intervals exists within the real numbers, thereby ensuring the existence and uniqueness of the intersection point when the interval lengths vanish.

Nested Interval Theorem

The Nested Interval Theorem is a classical result in real analysis which asserts that any sequence of nonempty, closed, nested intervals in the real numbers has a nonempty intersection. Moreover, if the lengths of the intervals tend to zero, the intersection consists of exactly one point. This theorem highlights the interplay between the structure of the real numbers and convergence properties.

Nested Intervals

Nested intervals refer to a sequence of intervals where each interval contains the next one. This hierarchical configuration is often used to 'zoom in' on a specific region or point by repeatedly restricting the domain, and is fundamental in arguments involving limits and convergence.

Closed Intervals

Closed intervals are sets of real numbers that include all the points between two endpoints, with the endpoints themselves included. This property is essential in analysis as it guarantees that limits of sequences within the interval remain in the interval, a feature that is exploited in various convergence proofs and theorems.

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