Question

Prove the following analogue of Theorem \(3.10(b):\) If \(\left\\{E_{a}\right\\}\) is a sequence of closed nonempty and bounded sets in a complete metric space \(X\), if \(E_{n} \supset E_{n+1}\), and if $$\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$$ then \(\cap_{1}^{\infty} E_{n}\) consists of exactly one point.

Short answer

Answer: In such a scenario, the intersection of all these sets consists of exactly one point.

Step-by-step solution

Define a Cauchy sequence

Since the sets form a nested sequence, we can choose a point from each set and form a sequence: \(x_n \in E_n\) for every \(n\). Now, because the limit of the diameters as \(n \to \infty\) is zero, for any \(\epsilon > 0\), there exists an \(N\) such that \(d(x_n, x_m) < \epsilon\), for all \(n, m > N\). Thus, the sequence \(\left\\{x_n\right\\}\) is a Cauchy sequence.

Identify a convergent point

Since the metric space \(X\) is complete, the Cauchy sequence \(\left\\{x_n\right\\}\) converges to a point, say \(x\). Due to the nested structure of the sets, we can write \(x_N \in E_1, E_2, \ldots, E_N\), and since every set is closed, we can conclude that \(x \in E_1, E_2, \ldots, E_N\). Hence, \(x \in \cap_{n=1}^{\infty} E_n\).

Show that the intersection consists of exactly one point

If the intersection consisted of more than one point, say \(x'\), then \(\operatorname{diam} E_n \ge d(x, x')\) for all \(n\). However, since \(\lim_{n \to \infty} \operatorname{diam} E_n = 0\), there exists an \(N\) such that \(\operatorname{diam} E_N < d(x, x')\). This contradicts our assumption, so \(x\) must be the only point in the intersection. Thus, we have proved that \(\cap_{1}^{\infty} E_n\) consists of exactly one point under the given conditions.

Key concepts

Complete Metric Space

A complete metric space is a key concept in understanding the behavior of sequences within metric spaces. Imagine a space where no sequence leaves you hanging—every sequence that should have a limit actually reaches it within the space. Mathematically, a metric space is considered complete if every Cauchy sequence converges to a point that is within the space. Think of it like having a conversation where every sentence that starts is guaranteed to finish within the room—a true sense of closure!

When dealing with nested sequences in metric spaces, the completeness property ensures that the sequences we're examining don’t just approach a point but actually reach a point within the space. This is crucial because it allows us to clearly define the limit point of sequences, which is a cornerstone in proving our textbook exercise.

Cauchy Sequence

The idea of a Cauchy sequence can be thought of as a promise of consistency within a sequence of points. In such a sequence, the points get arbitrarily close to each other as the sequence progresses. For any tiny distance you can think of, you'll eventually reach a stage in the sequence where all subsequent points are within that distance from each other.

In the context of our nested sequence problem, the Cauchy sequence comes about because the limit of the diameters goes to zero. This means that the distance between any two points in our sequence can be made as small as we like, by going far enough along the sequence. By forming a Cauchy sequence from the nested sets, and using the completeness of the space, we can confidently say that our sequence will converge to a point within the space.

Limit of Diameters

The limit of diameters going to zero is like ensuring that our closed sets are getting tighter and closer to trapping a single point without room to move. The diameter of a set refers to the greatest distance between any two points in the set. As we consider larger and larger indices in our nested sequence, the diameters shrink down to zero. This is pivotal because it means that, eventually, our sets become so small that they can’t possibly contain more than one point.

Within the nested sequence in our exercise, the diameter condition is the technical requirement that allows us to move from the idea of 'approaching' a point to actually 'having' a single point in the infinite intersection of sets, tying in beautifully with the concept of completions and Cauchy sequences.

Closed Set Property

A closed set is like a fence that contains its boundary. Any point that is snuggled up right against the edge of the set is still considered to be within the set. This contrasts with open sets, where points on the boundary are left out in the cold. The importance of being closed ties into our textbook exercise because, when a set is closed, any point that a sequence within the set converges to also belongs to the set.

This property is essential when proving the uniqueness of the point in the intersection of our nested sequence of sets. Through our exercise, once the Cauchy sequence finds its home at a point thanks to the complete metric space, the closed set property makes sure that this point is part of all the sets in our sequence, ensuring that none of the sets will betray our final result by excluding the point we've worked so hard to locate.