Question

Prove that if \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) are Cauchy sequences in \((S, \rho),\) then the sequence of distances $$ \rho\left(x_{m}, y_{m}\right), \quad m=1,2, \ldots $$ converges in \(E^{1}\).

Short answer

The sequence \(\rho(x_m, y_m)\) converges in \(E^1\).

Step-by-step solution

Understanding Cauchy Sequences

A sequence \(\{a_m\}\) in a metric space \((S, \rho)\) is Cauchy if for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n > N\), \(\rho(a_m, a_n) < \epsilon\). Cauchy sequences are characterized by the elements getting arbitrarily close to each other as they progress.

Verify Cauchy Condition for \(\{x_m\}\) and \(\{y_m\}\)

Given that \(\{x_m\}\) and \(\{y_m\}\) are Cauchy sequences, it means for any \(\epsilon > 0\), there exist integers \(N_1\) and \(N_2\) such that for all \(m, n > N_1\), \(\rho(x_m, x_n) < \frac{\epsilon}{2}\) and for all \(m, n > N_2\), \(\rho(y_m, y_n) < \frac{\epsilon}{2}\).

Sequence of Distances

We are interested in the sequence \(\rho(x_m, y_m)\) and whether it converges in \(E^1\). The goal is to show for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n > N\), \(|\rho(x_m, y_m) - \rho(x_n, y_n)| < \epsilon\).

Apply Triangle Inequality

Using the triangle inequality in the metric space \((S, \rho)\), we find:\[ \left| \rho(x_m, y_m) - \rho(x_n, y_n) \right| \leq \rho(x_m, x_n) + \rho(y_m, y_n). \] Since both sequences are Cauchy, we can bound each part as needed.

Bounding with Cauchy Condition

Select \(N = \max(N_1, N_2)\). For all \(m, n > N\), each term \(\rho(x_m, x_n) < \frac{\epsilon}{2}\) and \(\rho(y_m, y_n) < \frac{\epsilon}{2}\). Thus:\[ \left| \rho(x_m, y_m) - \rho(x_n, y_n) \right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. \] This shows the sequence \(\rho(x_m,y_m)\) is Cauchy, hence it converges.

Key concepts

Convergence

In the world of mathematics, the concept of convergence is crucial to understanding the behavior of sequences. A sequence converges if it approaches a particular value, known as the limit, as you progress through its terms.
In our exercise, the focus is on sequences of distances. If it can be shown that these distances converge, then they settle towards a specific number as the terms of the sequence increase.
The importance of convergence in sequences is that it allows us to make definitive statements about the large-scale behavior of sequences in mathematical analysis.

• Every Cauchy sequence in a metric space converges.
• The limit might be a number, like a rational or real number in Euclidean spaces, or a point in more abstract spaces.

Applying this principle, we ensure that the distances between pairs of terms from our two sequences \(\{x_m\}\) and \(\{y_m\}\) become negligible as \(m\) increases, driving towards convergence.

Metric Spaces

Metric spaces provide a formal way to capture and discuss distances within a set. More formally, a metric space is a set \(S\) along with a distance function \(\rho\), which assigns a non-negative real number indicating the distance between any two points in the set.
In our exercise, \(\rho(x, y)\) represents the distance between points \(x\) and \(y\) in the metric space \((S, \rho)\).
A key feature of metric spaces is that they lay the ground for determining properties such as convergence and Cauchy sequences.

• A sequence within a metric space can be analyzed through the distances between its points.
• The concept of distance helps in defining and verifying limits and convergence.

The characteristics of these spaces allow us to generalize many results from Euclidean spaces to more abstract settings, making them indispensable to advanced mathematics.

Triangle Inequality

An essential property of metric spaces and distances is the triangle inequality. This concept states that for any three points \(x, y, \) and \(z\) in a metric space, the direct distance between two points \(x\) and \(z\) is always less than or equal to the sum of distances from \(x\) to \(y\) and \(y\) to \(z\).
This principle can be captured mathematically as: \[\rho(x, z) \leq \rho(x, y) + \rho(y, z).\]
In the context of sequences, the triangle inequality supports bounding the differences between terms, which is crucial when demonstrating convergence for Cauchy sequences.
For our problem:

• Using the triangle inequality allows us to state \[|\rho(x_m, y_m) - \rho(x_n, y_n)| \leq \rho(x_m, x_n) + \rho(y_m, y_n).\]
• The inequality helps combine the Cauchy conditions of two separate sequences.

This provides a pathway to show that the sequence of distances eventually becomes negligible, ensuring it converges.