Question

Prove the sequential criterion for uniform continuity: A function \(f: A \rightarrow T\) is uniformly continuous on a set \(B \subseteq A\) iff for any two (not necessarily convergent) sequences \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) in \(B,\) with \(\rho\left(x_{m}, y_{m}\right) \rightarrow 0,\) we have \(\rho^{\prime}\left(f\left(x_{m}\right), f\left(y_{m}\right)\right) \rightarrow 0\) (i.e., \(f\) preserves concurrent pairs of sequences; see Problem 4 in Chapter \(3, \S 17 ).\)

Short answer

A function \( f: A \to T \) is uniformly continuous on \( B \subseteq A \) if and only if for sequences \( \{x_m\} \) and \( \{y_m\} \) in \( B \) with \( \rho(x_m, y_m) \to 0 \), we have \( \rho'(f(x_m), f(y_m)) \to 0 \).

Step-by-step solution

Understand Uniform Continuity

A function \( f: A \to T \) is uniformly continuous on \(B \subseteq A\) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x, y \in B \) with \( \rho(x, y) < \delta \), we have \( \rho'(f(x), f(y)) < \epsilon \).

Sequence Convergence Condition

We are given two sequences \( \{x_m\} \) and \( \{y_m\} \) in \( B \) such that \( \rho(x_m, y_m) \to 0 \). This means that for every \( \epsilon > 0 \), there exists an \( M \) such that for all \( m \geq M \), \( \rho(x_m, y_m) < \epsilon \).

Apply Uniform Continuity to Sequences

Since \( f \) is uniformly continuous on \( B \), there exists a \( \delta > 0 \) corresponding to any \( \epsilon' > 0 \) such that if \( \rho(x, y) < \delta \), then \( \rho'(f(x), f(y)) < \epsilon' \). For our sequences, choose \( \epsilon' > 0 \) and find the corresponding \( \delta > 0 \).

Prove the Sequence Limit Condition

For the sequences \( \{x_m\} \) and \( \{y_m\} \), since \( \rho(x_m, y_m) \to 0 \), there exists an \( M \) such that for all \( m \geq M \), \( \rho(x_m, y_m) < \delta \). By uniform continuity, this means \( \rho'(f(x_m), f(y_m)) < \epsilon' \) for all \( m \geq M \). Thus, \( \rho'(f(x_m), f(y_m)) \to 0 \).

Establish Equivalence of Conditions

The converse is also true: if \( \rho(x_m, y_m) \rightarrow 0 \) implies \( \rho^{\prime}(f(x_m), f(y_m)) \rightarrow 0 \) for any sequences \( \{x_m\} \) and \( \{y_m\} \), this ensures that for any given \( \epsilon > 0 \), choosing sequences \( \{x_m = x\} \) and \( \{y_m = y\} \) achieves \( \rho(x, y) < \delta \). This shows uniform continuity.

Key concepts

Sequential Criterion

The Sequential Criterion is a fascinating and useful mathematical concept when dealing with functions and their continuity properties. It provides a sequence-based method to determine uniform continuity. Instead of directly dealing with open intervals or neighborhoods, the Sequential Criterion allows us to express uniform continuity using sequences.

To apply it, consider sequences \( \{x_m\} \) and \( \{y_m\} \) coming from a set \( B \subseteq A \). When these sequences get closer—meaning the distance between them \( \rho(x_m, y_m) \) approaches zero—the corresponding outputs, according to the function \( f \), also ought to get closer. This is quantified by the condition \( \rho'(f(x_m), f(y_m)) \rightarrow 0 \).

If this criterion holds true for all possible pairs of sequences in \( B \), then \( f \) is uniformly continuous on set \( B \). Thus, this Sequential Criterion wraps the essence of uniform continuity into a sequence-oriented package.

Sequences in Analysis

In mathematical analysis, sequences are fundamental constructs used to explore the limits and behavior of functions. They are ordered lists of elements, typically numbers, and can offer critical insight into the nature of functions when individual elements of the sequence approximate arbitrary closeness.

Similar to the progression of numbers, sequences can exhibit various behaviors such as convergence, divergence, or oscillation. A common application in analysis is observing how sequences behave under functions. By examining the function's output sequences, one can infer continuity properties.

Engineers and mathematicians often use sequences to handle abstract problems or create simulations of real-world phenomena. Sequences thus serve as invaluable tools for both theoretical and applied mathematics, particularly when examining properties like uniform continuity.

Convergence of Sequences

A crucial concept in calculus and analysis is sequence convergence, which deals with the tendency of sequences to approach some limit. A sequence \( \{a_m\} \) is said to converge to a limit \( L \) if, for every positive number \( \epsilon \), however small, there exists a point in the sequence beyond which all terms are within \( \epsilon \) of \( L \).

Converging sequences are pivotal when discussing the continuity of functions, including uniform continuity. Uniform continuity can be considered a condition that is well-expressed through sequence behavior.

Use these properties to show a function \( f \) is uniformly continuous: if any two sequences \( \{x_m\} \) and \( \{y_m\} \) in a domain \( B \) converge to the same point, then the images under \( f \) converge to the same value—showing that a function preserves the convergence attribute of sequences.

• This can help ensure reliable function behavior over sets and is valuable in mathematical proofs.
• Ultimately, the convergence of sequences is a bridge to understanding deeper properties of functions, like uniform continuity.