Problem 3 - Basic Analysis: Introduction to Real Analysis

Question

Finish the proof of Proposition 2.3.6. That is, suppose \(\left\\{x_{n}\right\\}\) is a bounded sequence and \(\left\\{x_{n_{k}}\right\\}\) is a subsequence. Prove \(\liminf _{n \rightarrow \infty} x_{n} \leq \liminf _{k \rightarrow \infty} x_{n_{k}}\).

Step-by-Step Solution

Answer

Thus, \( \liminf_{n \to \infty} x_n \leq \liminf_{k \to \infty} x_{n_k} \).

Step 1: Understand the Definition of Lim Inf

The limit inferior (lim inf) of a sequence \( \{x_n\} \) as \( n \to \infty \) is defined as the greatest lower bound (infimum) of the set of all subsequential limits of \( \{x_n\} \). For the subsequence \( \{x_{n_k}\} \), the lim inf as \( k \to \infty \) is similarly defined. We need to show \( \liminf_{n \to \infty} x_n \leq \liminf_{k \to \infty} x_{n_k} \).

Step 2: Identify the Limit Inferior of the Subsequence

Consider the subsequence \( \{x_{n_k}\} \). Because \( \{x_{n_k}\} \) is a subsequence of \( \{x_n\} \), any limit point of \( \{x_{n_k}\} \) is also a limit point of \( \{x_n\} \). Define \( L = \liminf_{k \to \infty} x_{n_k} \). By definition, for any \( \varepsilon > 0 \), there exists a \( K \) such that for all \( k \geq K \), \( x_{n_k} > L - \varepsilon \).

Step 3: Connect the Sequence and Subsequences

Since \( x_{n_k} \) are terms of \( x_n \) itself, the values \( x_{n_k} > L - \varepsilon \) imply that there are infinitely many \( n \) such that \( x_n > L - \varepsilon \). Therefore, \( L - \varepsilon \) cannot be a strict upper bound on the values of \( x_n \) approaching lim inf. By definition of \( \liminf \), this implies \( \liminf_{n \to \infty} x_n \leq L = \liminf_{k \to \infty} x_{n_k} \).

Step 4: Formalize the Proof

Let \( \liminf_{n \to \infty} x_n = a \) and \( \liminf_{k \to \infty} x_{n_k} = L \). Assume for contradiction that \( a > L \). Then there exists \( \varepsilon > 0 \) such that \( a \geq L + \varepsilon \). The definition of \( a \) as the greatest lower bound implies all subsequential limits of \( x_n \) are at least \( a \). Hence, \( \liminf_{k \to \infty} x_{n_k} < a \), a contradiction. Therefore, \( a \leq L \).

Key Concepts

Limit Inferior

The concept of limit inferior, often written as \( \liminf_{n \rightarrow \infty} x_n \), is fundamental in real analysis. It provides insight into the behavior of a sequence as it progresses towards infinity.
Essentially, the limit inferior of a sequence \( \{x_n\} \) is the greatest lower bound (or infimum) of the set composed of all subsequential limits. It captures the lowest value that subsequences of \( \{x_n\} \) can converge to.

• This means that no subpart of the entire sequence can limit to a value lying below this bound.
• Understanding \( \liminf \) helps in analyzing the "floor" of subsequential behavior in extended sequences.

As such, in our proposition, we need to demonstrate \( \liminf_{n \to \infty} x_n \leq \liminf_{k \to \infty} x_{n_k} \) to understand how the sequence and its subsequences are intrinsically linked.

Subsequence

In real analysis, subsequences are key to revealing the hidden properties within sequences. A subsequence is derived by selecting elements from a sequence but in the same order as the original.
For instance, from a sequence \( \{x_n\} \), we can form a subsequence \( \{x_{n_k}\} \) by choosing indices that are strictly increasing. This means that each term \( x_{n_k} \) comes from \( \{x_n\} \) at positions determined stepwise from a selection process.

• Subsequences can provide insight into the behavior of the original sequence, especially when looking for convergence properties.
• Understanding subsequences is crucial as they form the building blocks for complex analysis in sequences.

Because subsequences inherit properties from their parent sequence, analyzing \( \{x_{n_k}\} \) unravels further information about \( \{x_n\} \), particularly in exploring convergence to particular limits, such as \( \liminf \).

Bounded Sequence

A bounded sequence is an essential type of sequence in real analysis where all terms lie within a fixed range. Specifically, a sequence \( \{x_n\} \) is bounded if there exists a real number \( M \) such that for all indices \( n \), \(|x_n| \leq M\).
This property ensures that the sequence doesn't deviate too wildly but instead remains within a predictable framework.

• Boundedness is critical because it can imply the existence of convergent subsequences; a result from the Bolzano-Weierstrass theorem.
• In practical terms, a bounded sequence assures us that the behavior of the sequence, and hence its subsequences, is controlled and predictable.

When addressing our proposition, the bounded nature of \( \{x_n\} \) guarantees that its behavior, especially involving limit infimum and subsequent subsequences, remains stable and analysable.

Subsequential Limits

Subsequential limits are the limiting points reached by subsequences of a given sequence \( \{x_n\} \). They represent the distinct values that various extracted subsequences can converge to.
Understanding subsequential limits is particularly valuable because not all sequences converge to a single limit. Instead, they might contain a variety of converging subtrajectories.

• Subsequential limits highlight the versatility and complexity within sequences; showcasing stability or oscillations.
• These limits are instrumental in determining the \( \liminf \) and \( \limsup \), as they contribute to finding the greatest lower or least upper bounds, respectively.

In our exercise, exploring the subsequential limits helps prove \( \liminf_{n \to \infty} x_n \leq \liminf_{k \to \infty} x_{n_k} \), by demonstrating how the sequence \( \{x_n\} \) and its subsequences \( \{x_{n_k}\} \) interact through their converging behaviors.