Question

Prove that \(\left\\{\left(x_{m}, y_{m}\right)\right\\}\) is a Cauchy sequence in \(X \times Y\) iff \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) are Cauchy. Deduce that \(X \times Y\) is complete iff \(X\) and \(Y\) are.

Short answer

\( \{(x_m, y_m)\} \) is Cauchy in \( X \times Y \) iff \( \{x_m\} \) and \( \{y_m\} \) are Cauchy. \( X \times Y \) is complete iff \( X \) and \( Y \) are complete.

Step-by-step solution

Understand the Definitions

A sequence \( \{(x_m, y_m)\} \) in \( X \times Y \) is a Cauchy sequence if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n, m > N \), \( d((x_n, y_n), (x_m, y_m)) < \epsilon \). Here, the distance in the product metric can be taken as \( \sqrt{d_X(x_n, x_m)^2 + d_Y(y_n, y_m)^2} \).

Show Sequence Conditions in Product Space

A sequence \( \{x_m\} \) is Cauchy in \( X \) if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n, m > N \), \( d_X(x_n, x_m) < \epsilon \). Similarly, \( \{y_m\} \) is Cauchy in \( Y \) if for all \( n, m > N \), \( d_Y(y_n, y_m) < \epsilon \).

Prove If Product Sequence is Cauchy Then Each is Cauchy

Assume \( \{ (x_m, y_m) \} \) is a Cauchy sequence in \( X \times Y \). For given \( \epsilon > 0 \), find \( N \) such that for all \( n, m > N \), \( \sqrt{d_X(x_n, x_m)^2 + d_Y(y_n, y_m)^2} < \epsilon \). This implies \( d_X(x_n, x_m) < \epsilon \) and \( d_Y(y_n, y_m) < \epsilon \), therefore \( \{ x_m \} \) is Cauchy in \( X \) and \( \{ y_m \} \) is Cauchy in \( Y \).

Prove If Each Sequence is Cauchy Then Product Sequence is Cauchy

Assume \( \{ x_m \} \) is Cauchy in \( X \) and \{ y_m \} is Cauchy in \( Y \). For given \( \epsilon > 0 \), exist \( N_1 \) and \( N_2 \) such that for all \( n, m > N_1 \), \( d_X(x_n, x_m) < \epsilon/\sqrt{2} \) and for all \( n, m > N_2 \), \( d_Y(y_n, y_m) < \epsilon/\sqrt{2} \). Let \( N = \max(N_1, N_2) \), then for all \( n, m > N \), the inequality \( \sqrt{d_X(x_n, x_m)^2 + d_Y(y_n, y_m)^2} < \epsilon \) also holds. Therefore, \( \{ (x_m, y_m) \} \) is Cauchy in \( X \times Y \).

Deduce Completeness Conditions

Since \( \{ (x_m, y_m) \} \) is Cauchy if and only if each component sequence is Cauchy, if \( X \) and \( Y \) are complete—every Cauchy sequence converges—then \( X \times Y \) is complete. Conversely, if \( X \times Y \) is complete, then any Cauchy sequence \( \{ (x_m, y_m) \} \) having convergent subsequences \( \{ x_m \} \), \{ y_m \} is complete, meaning \( X \) and \( Y \) must each be complete.

Key concepts

Metric Spaces

Metric spaces are fundamental in understanding the behavior of Cauchy sequences. A metric space includes a set of points and a notion of distance between any pair of points in that set. This distance is defined by a metric, which is a function that satisfies three key properties:

• Non-negativity: The distance between two points is always non-negative.
• Identity of indiscernibles: The distance between a point and itself is zero, while the distance between two distinct points is positive.
• Triangle inequality: The distance from one point to another is less than or equal to the sum of the distances through a third point.

In the context of the exercise, we're considering a product metric space, \( X \times Y \), meaning each point in this space is a pair consisting of an element from \( X \) and an element from \( Y \). The distance between two points \((x_n, y_n)\) and \((x_m, y_m)\) in this space is calculated using: \[ \sqrt{d_X(x_n, x_m)^2 + d_Y(y_n, y_m)^2} \].
This formula extends the idea of distance in individual metric spaces to pairs of points, adhering to the properties of a metric.

Completeness

Completeness is a property of a metric space where every Cauchy sequence converges to a limit within the same space. It's a crucial concept because it assures us that sequences not only come infinitely close to a certain point but also actually "reach" or equal that point within the bounds of the space.
A Cauchy sequence in a metric space \( X \) is characterized by the fact that, for any small positive distance, \( \epsilon \), the sequence's terms are eventually all closer to each other than \( \epsilon \). The completeness of a product metric space \( X \times Y \) requires that both individual metric spaces \( X \) and \( Y \) are complete.
This compatibility of completeness between spaces and their product ensures a consistent and predictable behavior of sequences when moving between and combining different spaces. If each sequence of the components converges within its respective space, then the combined sequence in the product space will also converge.

Convergence of Sequences

The convergence of sequences is a fundamental concept indicating that a sequence of points becomes arbitrarily close to some limit point as the sequence progresses towards infinity. For any Cauchy sequence in a complete metric space, convergence is guaranteed, meaning there's a point in the space (the limit) that the sequence approaches.
Cauchy sequences are inherently linked to convergence because they demonstrate that, beyond a certain point in the sequence, all subsequent elements are closer to each other, suggesting how their positions stabilize as the sequence unfolds.
In the case of this exercise, understanding the convergence of sequences in the context of product metric spaces involves recognizing that stability in each component sequence \( \{ x_m \} \) and \( \{ y_m \} \) in \( X \) and \( Y \) individually allow the whole sequence \( \{ (x_m, y_m) \} \) to converge in \( X \times Y \). Hence, the behavior of sequences in each metric informs the behavior in their product space, providing a comprehensive understanding of sequence convergence.