Question

Complete the details of the following alternative proof of Theorem \(4.19:\) If \(f\) is not uniformly continuous, then for some \(\varepsilon>0\) there are sequences \(\left\\{p_{n}\right\\},\left\\{q_{n}\right\\}\) in \(X\) such that \(d_{x}\left(p_{n}, q_{n}\right) \rightarrow 0\) but \(d_{r}\left(f\left(p_{n}\right), f\left(q_{n}\right)\right)>\varepsilon .\) Use Theorem \(2.37\) to obtain a contradiction.

Short answer

Answer: If a function f is not uniformly continuous, then for some ε > 0, there are sequences {p_n} and {q_n} in the space X such that the distance between p_n and q_n in the domain (d_x(p_n, q_n)) approaches 0, but the distance between f(p_n) and f(q_n) in the range (d_r(f(p_n), f(q_n))) is greater than ε.

Step-by-step solution

Assume f is not uniformly continuous

Since we're given that \(f\) is not uniformly continuous, there exists an \(\varepsilon > 0\) such that for all \(\delta > 0\), there exist points \(p, q \in X\) with \(d_x(p, q) < \delta\) and \(d_r(f(p), f(q)) \geq \varepsilon\).

Construct sequences \(p_n\) and \(q_n\)

For each natural number \(n\), choose \(p_n\) and \(q_n\) in \(X\) such that \(d_x(p_n, q_n) < \frac{1}{n}\) and \(d_r(f(p_n), f(q_n)) \geq \varepsilon\). This is possible because of the assumption that \(f\) is not uniformly continuous.

Show that \(d_x(p_n, q_n) \rightarrow 0\)

Since \(d_x(p_n, q_n) < \frac{1}{n}\) for all \(n\), and \(\frac{1}{n} \rightarrow 0\) as \(n \rightarrow \infty\), we have that \(d_x(p_n, q_n) \rightarrow 0\) by the squeeze theorem.

Apply Theorem 2.37

According to Theorem 2.37, if \(f\) were uniformly continuous, then for sequences \((p_n)\) and \((q_n)\) with \(d_x(p_n, q_n) \rightarrow 0\), we would have \(d_r(f(p_n), f(q_n)) \rightarrow 0\).

Reach a contradiction

However, we constructed the sequences \((p_n)\) and \((q_n)\) such that \(d_r(f(p_n), f(q_n)) \geq \varepsilon\) for all \(n\). This contradicts the implication of Theorem 2.37, and thus, the assumption that \(f\) is not uniformly continuous is indeed true. In conclusion, we have shown that if \(f\) is not uniformly continuous, then for some \(\varepsilon > 0\), there are sequences \(\left\\{p_{n}\right\\},\left\\{q_{n}\right\\}\) in \(X\) such that \(d_{x}\left(p_{n}, q_{n}\right) \rightarrow 0\), but \(d_{r}\left(f\left(p_{n}\right), f\left(q_{n}\right)\right)>\varepsilon\). This completes the alternative proof of Theorem 4.19.

Key concepts

Sequences

In mathematics, sequences are ordered lists of objects or numbers. Each number or object in a sequence is called a term. Sequences play a crucial role in understanding behaviors of functions and spaces. For example, in the exercise above, sequences \( \{p_n\} \) and \( \{q_n\} \) are constructed in a way that helps identify properties of the function \( f \).

A sequence can be finite or infinite, depending on the number of terms it includes. Infinite sequences continue indefinitely, which is often the case in calculus and real analysis. An important property that sequences can exhibit is convergence, where terms of the sequence approach a specific value as the sequence progresses.

• **Finite Sequence**: A sequence that has a limited number of terms.
• **Infinite Sequence**: A sequence with an unlimited number of terms.
• **Term (element)**: Each individual part or number in a sequence.

In our context, by constructing sequences \( \{p_n\} \) and \( \{q_n\} \) with terms getting arbitrarily close, we test the uniform continuity of the function \( f \). By carefully choosing sequences, we can effectively analyze the behavior of functions.

Convergence

Convergence describes how a sequence behaves as it progresses to infinity. Specifically, a sequence \((a_n)\) converges to a limit \(L\) if for every positive number \(\varepsilon\), there exists a natural number \(N\) such that for all \(n \geq N\), the terms \(a_n\) are within \(\varepsilon\) of \(L\). In simpler terms, the sequence gets closer and closer to \(L\) as you go further along.

The notion of convergence is key in areas like analysis, where it helps in understanding limits, continuity, and differentiability.

• **Limit**: The value a sequence approaches as the terms increase indefinitely.
• **Convergence Criterion**: \( |a_n - L| < \varepsilon \) for all \(n \geq N\).

In the provided exercise, the sequence \((d_x(p_n, q_n))\) converges to 0, illustrating that the points \(p_n\) and \(q_n\) get arbitrarily close. This aspect is crucial for arguing about the uniform continuity of \(f\), as a uniformly continuous function would imply that the distances \(d_r(f(p_n), f(q_n))\) would also converge to 0.

Contradiction Proof

Proof by contradiction is a standard mathematical technique used to establish the validity of a statement. The idea is to assume the opposite of what you want to prove, then show that this assumption leads to a logical contradiction. Therefore, the assumption must be false, and the original statement is shown to be true.

• **Assumption**: Start by assuming the negation of the statement you aim to prove.
• **Derivation of Contradiction**: Work through logical steps to reach a statement known to be false.
• **Conclusion**: Conclude the original statement must be true.

In our case, we assume that \(f\) is not uniformly continuous, and construct sequences that attempt to break this assumption. By leading to a contradiction with \( \varepsilon \), we substantiate that our original negation was false, thus proving the desired property of the function. This powerful technique is often used in mathematical proofs due to its straightforward and effective approach in dealing with complex problems.

Squeeze Theorem

The Squeeze Theorem is a fundamental principle in calculus and analysis, particularly helpful in proving the convergence of sequences. The theorem states that if you have three sequences \((a_n)\), \((b_n)\), and \((c_n)\) such that \(a_n \leq b_n \leq c_n\) for all \(n\), and \(a_n\) and \(c_n\) both converge to the same limit \(L\), then \((b_n)\) also converges to \(L\).

This theorem works by "squeezing" the sequence \((b_n)\) between two others that converge to the same point, forcing \((b_n)\) itself to converge to that point.

• **Bounding Sequences**: The sequences \((a_n)\) and \((c_n)\) that "trap" \((b_n)\).
• **Common Limit**: The shared limit \(L\) that \((a_n)\) and \((c_n)\) converge to.

In the exercise, we use the Squeeze Theorem to show that the distance \(d_x(p_n, q_n)\) converges to 0. Though \(d_x(p_n, q_n)\) is already naturally bounded by \(0\) and \(1/n\), it effectively illustrates the utility of the Squeeze Theorem in proving convergence within the context of mathematical analysis.