Problem 14 - Basic Analysis: Introduction to Real Analysis

Question

Find a convergent subsequence of the sequence \(\left\\{(-1)^{n}\right\\}\).

Step-by-Step Solution

Answer

Subsequences \(1, 1, 1, \ldots\) and \(-1, -1, -1, \ldots\) are convergent, to \(1\) and \(-1\) respectively.

Step 1: Understanding the Sequence

The sequence given is \((-1)^n\), which alternates between \(1\) and \(-1\). For even \(n\), it produces \(1\) (since \((-1)^{2k} = 1\)), and for odd \(n\), it produces \(-1\) (since \((-1)^{2k+1} = -1\)).

Step 2: Identifying Subsequences

A subsequence is derived by selecting some elements from the original sequence while maintaining the order. From \((-1)^n\), we can make subsequences by selecting elements corresponding to all the even index terms or odd index terms.

Step 3: Choosing the Convergent Subsequences

For even values of \(n\), let \(n = 2k\). Hence, the subsequence becomes \(\{1, 1, 1, \ldots\}\), and similarly, for odd \(n\), let \(n = 2k+1\). The subsequence becomes \(\{-1, -1, -1, \ldots\}\).

Step 4: Proving Convergence

Let us prove the convergence:1. Consider the subsequence \(1, 1, 1, \ldots\). As each term is \(1\), this subsequence converges to \(1\).2. Consider the subsequence \(-1, -1, -1, \ldots\). As each term is \(-1\), this subsequence converges to \(-1\).

Key Concepts

Subsequences

A subsequence is a concept in mathematics where we derive a new sequence from an original sequence by selecting certain elements and maintaining their original order. Think of it as picking certain terms from the list of a sequence, but you must keep them in the same order they appear in the original sequence. For example, if we consider the simple sequence of even numbers \( \{2, 4, 6, 8, \ldots\} \), a subsequence could be \( \{2, 6, \ldots\} \), where we just picked every other number. It's essential to note that subsequences can be infinite even if taken from an infinite sequence. This concept helps in analyzing the behavior of sequences, particularly in convergence where we look for parts of the sequence that tend towards a limit.

Sequence

A sequence is a list of numbers ordered in a specific manner where each number is called a term. In mathematics, sequences are functions whose domain is natural numbers. An example of a sequence is \( \{3, 6, 9, 12, \ldots\} \), known as the sequence of multiples of 3. Sequences can be either finite, meaning they have a last term, or infinite, with no end or last term.

• A sequence can be increasing, where each number is larger than the previous or decreasing, where each number is smaller.
• They can also be constant, like the sequence \( \{5, 5, 5, \ldots\} \), where all terms are identical.
• Another important property of sequences is their convergence, where a sequence approaches a particular value as the number of terms goes to infinity.

Understanding sequences and their behavior is crucial in various fields of mathematics and plays a vital role in real analysis.

Alternating Sequence

An alternating sequence is one where the terms alternate in sign or follow a pattern of variation. In our original problem, the sequence \((-1)^n\) alternates between 1 and -1 depending on whether \(n\) is even or odd. This gives us a sequence that looks like \(\{1, -1, 1, -1, \ldots\}\). Alternating sequences have interesting properties:

• They often do not converge, as there is no single value that all terms approach.
• However, they can contain convergent subsequences, as seen in our exercise where subsequences like \(\{1, 1, 1, \ldots\}\) converge to 1.

Alternating sequences are especially important when studying series and their convergence, as they can affect the overall behavior of a mathematical expression.

Real Analysis

Real analysis is a branch of mathematics dealing with real numbers and sequences and functions of real numbers. It involves the rigorous study of the properties of sequences, including their limits and convergence. The foundational concepts of real analysis include:

• Understanding limits and being able to determine when and how sequences converge.
• Delving into the behavior of functions, continuity, differentiation, and integration.
• Exploring the completeness of real numbers, which ensures that every Cauchy sequence of real numbers has a limit that is also a real number.

In real analysis, identifying convergent subsequences within a sequence, such as the those in the step-by-step solution given, is crucial for analyzing the overall behavior and properties of sequences, especially when they do not converge as a whole.