Question

If \(C_{1}, C_{2}, C_{3}, \ldots\) are sets such that \(C_{k} \supset C_{k+1}, k=1,2,3, \ldots, \lim _{k \rightarrow \infty} C_{k}\) is defined as the intersection \(C_{1} \cap C_{2} \cap C_{3} \cap \cdots .\) Find \(\lim _{k \rightarrow \infty} C_{k}\) if: (a) \(C_{k}=\\{x: 2-1 / k

Short answer

The limits for the series of sets are: (a)\(\{2\}\), (b)\(\{2\}\), (c)\(\{(0,0)\}\)

Step-by-step solution

Solve part (a)

The set defined in part (a), \(C_{k}=\{x: 2-1 / k<x \leq 2\}\), approaches the set \(\{2\}\) as \(k\) goes to infinity since the left limit \(2-1/k\) increases and approaches 2.

Solve part (b)

The set defined in part (b), \(C_{k}=\{x: 2<x \leq 2+1 / k\}\), approaches the set \(\{2\}\) as \(k\) goes to infinity since the right limit \(2+1/k\) decreases and approaches 2.

Solve part (c)

The set defined in part (c), \(C_{k}=\left\{(x, y): 0 \leq x^{2}+y^{2} \leq 1 / k\right\}\), represents the set of points inside a circle centered at the origin (0,0) with radius equal to square root of \(1/k\). As \(k\) goes to infinity, the radius of the circle goes to 0, which implies that the set \(C_{k}\) approaches the set \(\{(0,0)\}\), the point at the origin.

Key concepts

Convergence of Sets

When we talk about the convergence of a sequence of sets, we are referring to what the sets 'approach' as we move further and further along the sequence. Much like how we study the limit of a sequence of numbers, convergence in the context of sets involves a limit process. However, instead of numbers getting closer to a single value, convergence of sets is defined in terms of a sequence of sets becoming more focused or narrowing down on specific elements.

As seen in our textbook exercise, for a sequence of sets like \(C_1, C_2, C_3, \ldots\), the limit, denoted as \(\lim_{k \to \infty} C_k\), is the set that contains all elements that are in every set \(C_k\). This is technically the intersection of all sets in the sequence. In parts (a) and (b) of the exercise, as \(k\) increases, the defined ranges for \(x\) get smaller and converge to the single value 2, implying the sets are 'zeroing in' on this element. This illustrates the general idea of convergence for sequences of sets: they can narrow down to a single element or a smaller subset as \(k\) approaches infinity.

Understanding convergence is crucial, as it is used in various areas of mathematical analysis and probability theory, especially when dealing with events that occur over an infinite sequence of experiments or time periods.

Nested Sets

Nested sets are a special kind of sequence of sets where each set in the sequence contains the next set. It is as if the sets are placed within each other like a series of Russian dolls, with each doll being completely contained within the ones that came before it. In formal terms, a sequence of sets \(C_1, C_2, C_3, \ldots\) is said to be nested if for every \(k\), the following condition holds: \(C_{k} \supset C_{k+1}\).

In our exercise, we are given a perfect example of nested sets: each set \(C_{k}\) is contained in the previous set \(C_{k-1}\). This nesting property is valuable when applying the definition of the limit of a sequence of sets. Because the sets are nested, their limit is simply the intersection of all sets in the sequence. This is particularly helpful in part (c), where each subsequent set represents a smaller concentric circle around the origin, thereby nesting within one another.

Understanding nested sets is key when dealing with sequences where order and inclusion are vital, as it often simplifies the approach to finding limits of complex set sequences.

Infinitely Decreasing Sets

An infinitely decreasing sequence of sets is a type of nested sequence where each set is not only contained within its predecessor but also shrinks without bound as the sequence progresses. As the name suggests, such sequences decrease infinitely, moving towards a smaller set or, in some cases, becoming empty.

This concept appears vividly in part (c) of our exercise, where each set \(C_{k}\) is a circle with a radius that decreases as \(k\) increases, suggesting that the circles become infinitesimally small. As \(k\) approaches infinity, these sets shrink towards a single point, the origin \((0,0)\). Essentially, the sequence of sets 'collapses' down to a point.

Infinitely decreasing sets have various applications, including geometric interpretations, like the circles in our exercise, and in defining probabilities in the case of events that become less probable over time. Understanding such sets is vital for grasping concepts of convergence in more abstract metric spaces and measure theory, where notions of 'size' and 'distance' require careful definition and examination.