Question

Prove that a sequence \(\left\\{x_{m}\right\\}\) is Cauchy in \((S, \rho)\) iff $$ (\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho\left(x_{m}, x_{k}\right)<\varepsilon $$

Short answer

The sequence is Cauchy iff \((\forall \varepsilon>0)(\exists k)(\forall m>k) \rho(x_m, x_k)<\varepsilon\).

Step-by-step solution

Understanding Cauchy Sequences

A sequence \(\{x_m\}\) is called a Cauchy sequence in a metric space \((S, \rho)\) if for every \(\varepsilon > 0\), there exists an integer \(N\) such that for all integers \(m, n > N\), \(\rho(x_m, x_n) < \varepsilon\). This means that the terms of the sequence become arbitrarily close as the sequence progresses.

Reformulating Condition for Cauchy Sequence

We need to show that \((\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho\left(x_{m}, x_{k}\right)<\varepsilon\)\ is equivalent to the Cauchy condition. This involves expressing the behavior of the sequence such that each term past a certain index \(k\) is close to the term at \(k\), with respect to \(\varepsilon\).

Proving Forward Implication

Assume \((\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho(x_m, x_k) < \varepsilon\). For given \(\varepsilon>0\), this provides an index \(k\) where for all \(m>k\), \(\rho(x_m, x_k) < \varepsilon/2\). Now, for any \(n > m > k\), by the triangle inequality, \(\rho(x_n, x_m) \leq \rho(x_n, x_k) + \rho(x_m, x_k) < \varepsilon/2 + \varepsilon/2 = \varepsilon\), satisfying the Cauchy definition.

Proving Reverse Implication

Assume the sequence is Cauchy. Thus, for \(\varepsilon>0\), there exists \(N \) such that for \(m, n > N\), \(\rho(x_m, x_n) < \varepsilon\). Now, set \(k = N\). For any \(m > k\), \(\rho(x_m, x_k) < \varepsilon\), proving \((\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho(x_m, x_k) < \varepsilon\).

Key concepts

Understanding Metric Spaces

A metric space is a fundamental concept in mathematics, especially in analysis. It provides a way to describe the space in which sequences, like those discussed in the exercise, can be analyzed based on distance. A metric space consists of a set \( S \) together with a function \( \rho \) (called a metric or distance function) that defines the distance between any two elements in \( S \). This metric function \( \rho \) must satisfy the following properties:

• Non-negativity: \( \rho(x, y) \geq 0 \) for all \( x, y \in S \), and \( \rho(x, y) = 0 \) if and only if \( x = y \).
• Symmetry: \( \rho(x, y) = \rho(y, x) \) for all \( x, y \in S \).
• Triangle inequality: \( \rho(x, y) + \rho(y, z) \geq \rho(x, z) \) for all \( x, y, z \in S \).

These properties ensure that the metric space behaves like our intuitive understanding of distance, making it a valuable tool for analyzing Cauchy sequences.

Sequence Convergence Essentials

Sequence convergence is an essential concept to understand in the context of metric spaces. It describes how a sequence behaves as it progresses and whether it approaches a particular value. A sequence \( \{a_n\} \) in a metric space \( (S, \rho) \) is said to converge to a limit \( L \in S \) if for every positive number \( \varepsilon \), no matter how small, there exists an integer \( N \) such that for all \( n > N \), the distance between \( a_n \) and \( L \) is less than \( \varepsilon \).
In simpler terms:

• As \( n \) increases, \( a_n \) gets arbitrarily close to \( L \).
• The smaller the \( \varepsilon \), the sooner \( a_n \) terms must stay within the distance \( \varepsilon \) of \( L \).

This idea is crucial for understanding when and how sequences "settle down" to a specific point in a metric space. Cauchy sequences, in particular, often feature in discussions about convergence.

Exploring the Triangle Inequality

The triangle inequality is a key property of metric spaces that plays an important role in many theorems, including those involving Cauchy sequences. This property states that for any three elements \( x, y, z \) in a metric space \((S, \rho)\), the following inequality holds: \[\rho(x, z) \leq \rho(x, y) + \rho(y, z)\]
This principle suggests that the direct path between two points is always the shortest compared to any roundabout path involving an intermediate point.

In the context of proving whether a sequence is Cauchy, the triangle inequality helps to establish upper bounds for distances. For example, if you know \( \rho(x_n, x_k) < \varepsilon/2 \) and \( \rho(x_m, x_k) < \varepsilon/2 \), then for \( x_n \) and \( x_m \):

• The inequality gives \( \rho(x_n, x_m) \leq \rho(x_n, x_k) + \rho(x_m, x_k) < \varepsilon \).

Thus, the triangle inequality is instrumental in analyzing and proving the properties of sequences in metric spaces.

Understanding the Limit of a Sequence

The concept of the limit of a sequence is central in calculus and analysis. It describes the value that the terms of a sequence \( \{a_n\} \) approach as \( n \) becomes very large. When a sequence has a limit \( L \), it means that as \( n \) increases, the terms \( a_n \) get arbitrarily close to \( L \).
To mathematically express that \( \{a_n\} \) converges to \( L \):

• For every \( \varepsilon > 0 \), there exists an integer \( N \) such that for all \( n > N \), the distance \( \rho(a_n, L) < \varepsilon \).

This condition describes how within some distance threshold \( \varepsilon \), the sequence remains close to the limit \( L \) after a certain point in the sequence.

The limit of a sequence is crucial in defining a Cauchy sequence's behavior. In metric spaces, a Cauchy sequence is one that is pre-supposed to be converging to some limit (although sometimes the limit may not be apparent within the space unless it is complete). Understanding limits helps identify how a sequence "settles down" to a specific value.