Question

Show that the following sequences of sets, \(\left\\{C_{k}\right\\}\), are nondecreasing, (1.2.16), then find \(\lim _{k \rightarrow \infty} C_{k}\). (a) \(C_{k}=\\{x: 1 / k \leq x \leq 3-1 / k\\}, k=1,2,3, \ldots\). (b) \(C_{k}=\left\\{(x, y): 1 / k \leq x^{2}+y^{2} \leq 4-1 / k\right\\}, k=1,2,3, \ldots\)

Short answer

Both sequences of sets \(\{C_k\}\) are nondecreasing. The limit as \(k\) approaches infinity for sequence (a) is given by \(\{x: 0 \leq x \leq 3\}\) and for sequence (b) is given by \(\{(x, y):0 \leq x^{2}+y^{2} \leq 4\}\).

Step-by-step solution

Nondecreasing Proof for Part (a)

For \(C_k = \{x: 1 / k \leq x \leq 3-1 / k\}\), note that as \(k\) increases, \(\frac{1}{k}\) decreases and \(3-\frac{1}{k}\) also decreases. Therefore, every \(C_k\) is a subset of \(C_{k+1}\), so the sequence is nondecreasing.

Find the Limit for Part (a)

The limit as \(k\) approaches infinity can be found by noting that as \(k\) approaches infinity, \(1 / k\) approaches 0 and \(3-1 / k\) approaches 3. Therefore, the union over all \(C_k\)'s which would be \(lim _{k \rightarrow \infty} C_{k} = \{x: 0 \leq x \leq 3\}\).

Nondecreasing Proof for Part (b)

For \(C_k =\{(x, y): 1 / k \leq x^{2}+y^{2} \leq 4-1 / k\}\), as \(k\) increases, \(1 / k\) decreases and \(4-1/k\) also decreases. Therefore, the intersection of \(x^{2}+y^{2}\) with the decreasing interval also decreases, so every \(C_k\) is a subset of \(C_{k+1}\), and the sequence is nondecreasing.

Find the Limit for Part (b)

The limit as \(k\) approaches infinity can be found by noting that as \(k\) approaches infinity, \(1/k\) approaches 0, and \(4-1/k\) approaches 4. Therefore, the union over all \(C_k\)'s which would be \(lim _{k \rightarrow \infty} C_{k} =\{(x, y):0 \leq x^{2}+y^{2} \leq 4\}\).

Key concepts

Limit of a Sequence

Understanding the limit of a sequence is foundational in calculus and analysis. It's the value that the elements of a sequence get infinitely close to as the index increases without bounds. Imagine a sequence of numbers getting closer and closer to a single number; that single number is the limit.

In our exercise with the set \(C_k = \{x: 1 / k \leq x \leq 3-1 / k\}\), we see that as \(k\) goes to infinity, the values of \(1/k\) and \(3-1/k\) approach 0 and 3, respectively. Hence, the limit of this sequence is the closed interval \[0,3\]. The concept of sequence limits extends beyond numbers to include sets and functions, yielding profound implications in continuous mathematics.

Set Theory

At the heart of modern mathematics lies set theory, a branch that studies sets or collections of objects. It's the foundation for various mathematical disciplines, including analysis, which deals with sequences and their limits. In our exercise, the sets \(C_k\) are collections of real numbers defined by certain conditions.

A key concept in set theory is the idea of a sequence of sets being nondecreasing, which means every set in the sequence is a subset of the ones following it. This property is crucial when deducing the limit of a sequence of sets, as it guarantees that there is a well-defined union (limit) of all sets in the sequence.

Real Number Intervals

The concept of real number intervals is vital in understanding continuous ranges in mathematics. Intervals can be open, closed, or half-open, and they represent all numbers lying between a set of endpoints. The difference between the types of intervals lies in whether they include their endpoints or not.

For instance, in the sequence \(C_k\) from the exercise, each \(C_k\) represents a closed interval since it includes its endpoints, \(1/k\) and \(3-1/k\). As \(k\) becomes larger, these endpoints converge, and we examine the development of these intervals to understand the limit of the sequence.

Mathematical Proofs

A mathematical proof is a logical argument that establishes the truth of a mathematical statement. Proofs use definitions, axioms, previously established statements, and logical deductive reasoning to arrive at a conclusion. In the context of our exercise, the proof shows that the sequences \(C_k\) are nondecreasing.

The proof consists of demonstrating that for each integer \(k\), the set \(C_k\) is a subset of \(C_{k+1}\), which conveys the nondecreasing nature of the sequence. Through mathematical proof, we can rigorously confirm the behavior of the sequence and its limit, adding certainty and understanding to our mathematical worldview.