Question

By the diameter of a subset \(A\) of a metric space \(R\) is meant the number $$ d(A)=\sup _{x, y \in A} p(x, y) $$ Suppose \(R\) is complete, and let \(\\{A,\), be a sequence of closed subsets of \(R\) nested in the sense that $$ A_{1} \supset A_{2} \supset \ldots \supset A_{n} \supset \cdots $$ Suppose further that $$ \lim _{n \rightarrow \infty} d\left(A_{n}\right)=0 $$ Prove that the intersection \(\bigcap_{n=1}^{\sim} A\), is nonempty.

Short answer

The intersection \( \bigcap_{n=1}^{\infty} A_n \) is nonempty due to nested property and diameter convergence in a complete metric space.

Step-by-step solution

Understanding the Problem

We have a sequence of nested closed subsets \( A_n \) in a complete metric space \( R \). Each subset has a diameter \( d(A_n) \) which approaches zero as \( n \) goes to infinity. We are tasked to show that the intersection \( \bigcap_{n=1}^{\infty} A_n \) is nonempty.

Analyze Diameter Convergence

The diameter of a set \( A \), \( d(A) = \sup_{x,y \in A} p(x, y) \), represents the greatest distance between any two points in \( A \). As \( n \to \infty \), \( d(A_n) \to 0 \), implying that points within \( A_n \) must get arbitrarily close to each other as \( n \) increases.

Use Completeness of the Space

The completeness of \( R \) implies that every Cauchy sequence in \( R \) has a limit that is also in \( R \). Since \( d(A_n) \to 0 \), sequences formed by choosing points in each \( A_n \) are Cauchy sequences.

Construct a Cauchy Sequence

Select a point \( x_n \) from each \( A_n \). The sequence \( (x_n) \) is a Cauchy sequence because these points get arbitrarily close due to \( d(A_n) \to 0 \).

Find the Limit of the Cauchy Sequence

Since \( (x_n) \) is a Cauchy sequence in the complete space \( R \), it has a limit point, say \( x \). This point \( x \) must belong to every \( A_n \) as \( \{A_n\} \) are nested, closed subsets.

Conclude the Intersection is Nonempty

Since the limit point \( x \) belongs to every \( A_n \), it also belongs to the intersection \( \bigcap_{n=1}^{\infty} A_n \). Therefore, the intersection is nonempty.

Key concepts

Diameter of a Set

The diameter of a set, in the context of a metric space, provides insight into how 'spread out' the points within the set are. For any subset \(A\) of a metric space \(R\), the diameter \(d(A)\) is defined as the supremum of the distances between any pair of points \(x\) and \(y\) in \(A\). In simpler terms, it is the largest possible distance you can find between any two points within the subset. This can be expressed mathematically as:
\[ d(A) = \sup_{x, y \in A} p(x, y) \]
This concept becomes particularly useful when analyzing sequences of sets that are shrinking, where the diameter approaching zero can suggest that the points within these sets are getting closer together. The diameter shrinking to zero is a strong hint that the sets are converging to a single point or a set of points that are very close to each other.

Nested Subsets

Nested subsets are like a set of Russian dolls; each subset in the sequence is contained within the one before it. Formally, for a sequence of sets \(A_n\), we say they are nested if:\[ A_1 \supset A_2 \supset A_3 \supset \ldots \supset A_n \supset \ldots \]
Nesting is an important concept in analysis because it provides order and structure to the sequence of sets, allowing us to make predictions about their behavior. Closed nested subsets, especially when found in complete metric spaces, can have very interesting properties. When working in a complete metric space, if the nested subsets also have diameters that shrink to zero, it ensures there's at least one point that is contained within all of them. This comes from the completeness of the space and the closure of the sets, giving rise to nonempty intersections even as the subsets become very small.

Cauchy Sequence

A Cauchy sequence is a key concept when discussing convergence and completeness in metric spaces. It is a sequence where the elements get arbitrarily close to each other as the sequence progresses. Formally, a sequence \((x_n)\) is called Cauchy if, for every positive number \(\epsilon\), there exists an index \(N\) such that for all \(m, n > N\), the distance \(p(x_m, x_n) < \epsilon\). This means that eventually all the terms in the sequence cluster together very tightly.

• A Cauchy sequence does not need to have a limit within the space unless the space is complete.
• Completeness is the property that assures these sequences do indeed have a limit in the same space.

In the context of nested subsets with shrinking diameters, selecting points from each subset can form a Cauchy sequence. Since the entire space is complete, this Cauchy sequence will converge to a point within the space, which must belong to the intersection of all subsets. This guarantees that this intersection is nonempty.