Question

Prove that if \(\left\\{f_{n}\right\\}\) tends to \(f\) (pointwise or uniformly), so does each subsequence \(\left\\{f_{n_{k}}\right\\}\).

Short answer

Each subsequence of a convergent sequence converges to the same limit.

Step-by-step solution

Define Sequence Convergence

To show that the subsequence \( \{f_{n_k}\} \) converges to \( f \), we begin by recalling the definition of convergence for sequences. A sequence \( \{f_n\} \) converges to a function \( f \) pointwise if for every \( x \) and for every \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n \geq N \), \( |f_n(x) - f(x)| < \epsilon \). For uniform convergence, this condition must hold uniformly for all \( x \).

Subsequence Inheritance of Limits

A subsequence inherits the convergence properties of its parent sequence. Since \( \{f_n\} \to f \), for a given \( \epsilon > 0 \), there is an \( N \) such that \( |f_n(x) - f(x)| < \epsilon \) for all \( n \geq N \), and this condition holds for the subsequence \( \{f_{n_k}\} \) because \( \{n_k\} \) is a subset of the indices \( n \geq N \).

Applying Convergence to Subsequence

Since the subsequence \( \{f_{n_k}\} \) consists of terms from the parent sequence \( \{f_n\} \) that already satisfy the convergence criteria by Step 2, we have for each \( x \) and \( \epsilon > 0 \), there exists the same \( N \) ensuring \( |f_{n_k}(x) - f(x)| < \epsilon \) for all \( n_k \geq N \).

Conclude the Proof

As we have shown that the subsequence \( \{f_{n_k}\} \) satisfies the necessary convergence criteria for both pointwise and uniform convergence, it follows from Steps 1-3 that \( \{f_{n_k}\} \to f \). Thus, any subsequence of a convergent sequence to a limit \( f \) will also converge to \( f \).

Key concepts

Pointwise Convergence

Pointwise convergence is a concept applied to sequences of functions where convergence occurs at each point individually in the domain. Suppose we have a sequence of functions \( \{f_n\} \) converging to a function \( f \). For pointwise convergence, we assert that for every point \( x \), the function values \( f_n(x) \) get arbitrarily close to \( f(x) \) as \( n \) becomes large.
To make this precise: for each \( x \) and any \( \epsilon > 0 \), there exists a number \( N \) such that for all \( n \ge N \), the condition \( |f_n(x) - f(x)| < \epsilon \) holds true. This definition emphasizes checking convergence at each individual point \( x \).
Essentially, while dealing with pointwise convergence, you're focusing on specific points in the function's domain. It doesn't necessarily say anything about how the functions behave as a complete entity across this domain. Each point can converge at different rates or even behave differently across the sequence which sometimes fails to capture the overall behavior of function sequences.

Uniform Convergence

Uniform convergence is a stronger form of convergence compared to pointwise convergence. When a sequence of functions \( \{f_n\} \) converges uniformly to a function \( f \), it means that the convergence occurs simultaneously across the entire domain.
Uniform convergence requires that for every \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n \ge N \) and for every point \( x \) in the domain, \( |f_n(x) - f(x)| < \epsilon \). This condition has to be satisfied for every \( x \) and is independent of \( x \).

• Uniform convergence ensures continuity: If each \( f_n \) is continuous and \( \{f_n\} \) converges uniformly, the limit function \( f \) will also be continuous.
• It is often more desirable than pointwise convergence because it preserves the overall structure and properties of functions across the entire domain.

With uniform convergence, you can think of the sequence converging to \( f \) like a group moving together, maintaining consistency at all points rather than individually at selected spots.

Sequence Convergence

Sequence convergence refers broadly to the idea that elements within a sequence are approaching a specific value as the sequence progresses. For a sequence of numbers \( \{a_n\} \), convergence to some number \( L \) implies that for any \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n \ge N \), the condition \( |a_n - L| < \epsilon \) must hold.
In the context of functions, this concept extends to sequence convergence of function values approaching a limiting function \( f \).

• In mathematical analysis, such sequences are crucial as they establish the foundation for limits, continuity, and differentiability. Consequently, examining subsequences, consisting of elements extracted from an original sequence while preserving their order, is important in understanding the broader behavior of sequences.
• Knowing that every convergent sequence yields convergent subsequences is pivotal. This property alleviates the proof process in problems dealing with limits, as any subsequence inheriting this convergence will converge to the same limit as the original sequence.

Convergence of sequences, whether numerical or functional, acts as a cornerstone of mathematical analysis, offering insights into the behavior and properties of functions or numbers within those sequences.