Question

Construct a sequence \(\left\\{s_{n}\right\\}\) with the following property, or show that none exists: for each positive integer \(m,\left\\{s_{n}\right\\}\) has a subsequence converging to \(m\).

Short answer

Question: Construct a sequence \(\left\\{s_{n}\right\\}\) such that for every positive integer \(m\), it has a subsequence converging to \(m\), or show why it doesn't exist. Answer: Consider the sequence \(s_1=1, s_2=1, s_3=2, s_4=1, s_5=2, s_6=3, s_7=1, s_8=2, s_9=3, s_{10}=4, \dots\). This sequence repeats the numbers from \(1\) to \(n\) once, where \(n\) keeps increasing. As every positive integer \(m\) is included in the sequence multiple times, we can find subsequences converging to \(m\). Therefore, such a sequence exists.

Step-by-step solution

Understanding the problem

From the given problem, we know that for every positive integer \(m\), there must exist a subsequence of \(\left\\{s_{n}\right\\}\) converging to \(m\). This means our sequence must cover all positive integers infinitely many times to ensure we can find these convergent subsequences. Step 2: Constructing the sequence

Construct a sequence with required properties

Consider the following sequence: \(s_1=1, s_2=1, s_3=2, s_4=1, s_5=2, s_6=3, s_7=1, s_8=2, s_9=3, s_{10}=4, \dots\) This sequence is constructed in the following manner: - Include the number \(1\) once - Repeat the numbers \(1\) and \(2\) once - Repeat the numbers from \(1\) to \(3\) once - Repeat the numbers from \(1\) to \(4\) once - Continue this pattern, repeating the numbers from \(1\) to \(n\) once, where \(n\) keeps increasing. Step 3: Checking the properties of the constructed sequence

Verify if the sequence fulfills the required properties

From the construction of the sequence, we can see that every positive integer \(m\) is included in the sequence multiple times. We can find subsequences that only take elements from one of those repetitions, and these subsequences will converge to \(m\). For example: - For \(m=1\), there is a subsequence formed by taking the first element of each repetition which is \(1,1,1,\dots\), converging to \(1\). - For \(m=2\), there is a subsequence formed by taking the last element of the first repetition (2), and then the last element of the second repetition (2), and so on, which is \(2,2,2,\dots\), converging to \(2\). - And we can continue this pattern for every positive integer \(m\). Thus, we have constructed a sequence \(\left\\{s_{n}\right\\}\) that has a subsequence converging to every positive integer \(m\).

Key concepts

Subsequence

In the realm of sequences and series, a subsequence is a new sequence that you can derive by removing some elements of the original sequence without changing the order of the remaining elements. Imagine you have a long list of numbers; a subsequence is like taking a walk through this list and picking some numbers along the way while maintaining the order they appeared in.
A classic example is a sequence of numbers: if the sequence is 2, 4, 6, 8, 10, then a subsequence could be 4, 8, 10 because they appear in the same order as in the original list. This concept is essential in analysis as it helps in studying the convergence properties and limits of sequences.
For any given sequence, infinite possible subsequences can be constructed, and each one may have different convergence properties. Some subsequences might converge to specific values, while others may not converge at all. By analyzing subsequences, especially those that converge, we can better understand the behavior of the sequence as a whole.

Positive Integers

Positive integers are the set of numbers starting from 1 and going upwards: 1, 2, 3, and so on, endlessly. They do not include zero, negative numbers, or any fractional components. They are foundational in mathematics and often considered the basic building blocks for much of arithmetic and number theory.
In the context of the original exercise, positive integers play a crucial role as they are the target values we want our sequence \(\{s_{n}\}\) to converge to. To construct a sequence that has subsequences converging to any positive integer means ensuring that the sequence includes each positive integer multiple times, in a pattern that allows selection of subsequences. By doing so, we can represent infinitely many parts of our sequence as statements like 1, 1, 1, or 2, 2, 2, converging clearly to that integer.
Understanding positive integers helps us appreciate their discrete and simple structure, providing clarity when constructing sequences or performing operations in proofs and real analysis.

Real Analysis

Real analysis is a branch of mathematics dealing with real numbers and the analytic properties of real-valued sequences and functions. It involves rigorously developing the theories of calculus using the real number system and other related structures like sequences and limits.
The field is fundamental for exploring how functions behave in continuous spaces, how sequences converge, and the rigorous definitions of limits and continuity. It delves into related concepts such as differentiation, integration, and infinite series, laying the groundwork for much of modern mathematical analysis.
In the given exercise, real analysis helps in understanding how sequences behave when they are infinitely long and how they can converge to particular values. The exercise involves showing that for every positive integer, there exists a subsequence converging to that integer, requiring insight into the structure and ordering of real numbers, as well as sequences. This endeavor involves applying the principles of limits and convergence, core topics in real analysis, to construct and justify the behavior of the sequences involved.