Problem 10 - Basic Analysis: Introduction to Real Analysis

Question

Take \(\mathbb{R}^{*}=\\{-\infty\\} \cup \mathbb{R} \cup\\{\infty\\}\) be the extended reals. Define \(d(x, y):=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right|\) if \(x, y \in \mathbb{R},\) define \(d(\infty, x):=\left|1-\frac{x}{1+|x|}\right|, d(-\infty, x):=\left|1+\frac{x}{1+|x|}\right|\) for all \(x \in \mathbb{R},\) and let \(d(\infty,-\infty):=2\) a) Show that \(\left(\mathbb{R}^{*}, d\right)\) is a metric space. b) Suppose \(\left\\{x_{n}\right\\}\) is a sequence of real numbers such that for every \(M \in \mathbb{R},\) there exists an \(N\) such that \(x_{n} \geq M\) for all \(n \geq N .\) Show that \(\lim x_{n}=\infty\) in \(\left(\mathbb{R}^{*}, d\right)\) c) Show that a sequence of real numbers converges to a real number in \(\left(\mathbb{R}^{*}, d\right)\) if and only if it converges in \(\mathbb{R}\) with the standard metric.

Step-by-Step Solution

Answer

\((\mathbb{R}^{*}, d)\) is a metric space; a sequence converging to infinity in \(\mathbb{R}^{*}\) behaves as expected, and real convergence holds true under both metrics.

Step 1: Show Non-negativity

To prove that \( (\mathbb{R}^{*}, d) \) is a metric space, we first need to show that \( d(x, y) \geq 0 \) for all \( x, y \in \mathbb{R}^{*} \). This is evident because the absolute value function \( |\cdot| \) is non-negative.

Step 2: Show Identity Property

We need to prove that \( d(x, y) = 0 \) if and only if \( x = y \). For \( x, y \in \mathbb{R} \), \( d(x, y) = \left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right| = 0 \) holds if and only if \( x = y \). Similarly, for infinity, \( d(\infty, x) = |1 - \frac{x}{1+|x|}| = 0 \) if and only if \( x = \infty \), and for negative infinity, \( d(-\infty, x) = |1 + \frac{x}{1+|x|}| = 0 \) if and only if \( x = -\infty \). Thus, the identity property is satisfied.

Step 3: Show Symmetry

The distance function \( d(x, y) \) must be symmetric, meaning \( d(x, y) = d(y, x) \). This holds since \( \left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right| = \left|\frac{y}{1+|y|}-\frac{x}{1+|x|}\right| \). This symmetry condition is trivially satisfied when dealing with infinities as well.

Step 4: Show Triangle Inequality

To satisfy the triangle inequality:\[ d(x, z) \leq d(x, y) + d(y, z) \]for any \( x, y, z \in \mathbb{R}^{*} \). For real numbers, this can be proven by using the property of absolute values: \( |a - c| \leq |a - b| + |b - c| \). Similar arguments hold for infinities using properties of bounded distances (like \( d(\infty, x) + d(x, \infty) \)).

Step 5: Prove Convergence to Infinity

Let \( \{x_n\} \) be a sequence such that for any \( M \), there exists \( N \) where \( x_n \geq M \) for all \( n \geq N \). Here, \( d(\infty, x_n) = |1 - \frac{x_n}{1+|x_n|}| \) tends to \( 0 \) as \( x_n \to \infty \), fulfilling the definition of \( x_n \to \infty \) in \( (\mathbb{R}^*, d) \).

Step 6: Convergence in Extended Reals

We need to show that a sequence converges to a real number \( a \) in \( (\mathbb{R}^*, d) \) if it converges to \( a \) in \( \mathbb{R} \) with the standard metric. If \( \{x_n\} \) converges to \( a \) with the standard metric, the definition naturally leads \( \{ \frac{x_n}{1+|x_n|} \} \) to tend towards \( \frac{a}{1+|a|} \), thus \( d(x_n, a) \to 0 \) in the extended reals.

Key Concepts

Metric Space

In mathematics, a metric space is a set where a distance function, known as a metric, defines the distance between any two points in the set. This concept is crucial because it allows us to talk about the closeness of elements, how they might converge, or how they might relate spatially.
To be considered a metric space, the distance function must satisfy four key properties:

• Non-negativity: For any two points, the distance between them must be greater than or equal to zero, i.e. \(d(x, y) \geq 0\).
• Identity of Indiscernibles: The distance between two points is zero if and only if the points are exactly the same, \(d(x, y) = 0\) iff \(x = y\).
• Symmetry: The distance from point A to point B is the same as from point B to A, thus, \(d(x, y) = d(y, x)\).
• Triangle Inequality: This asserts that the direct distance between two points should be less than or equal to taking a detour through an intermediate point, \(d(x, z) \leq d(x, y) + d(y, z)\).

These properties make metric spaces highly versatile in studying the structure of sets, including topics such as convergence, continuity, and compactness in real analysis.

Extended Real Numbers

The extended real number system is an extension of the usual real numbers \(\mathbb{R}\) that includes two additional elements: positive infinity (\( +\infty\) ) and negative infinity (\( -\infty\) ). This system is particularly useful in real analysis for discussing limits and improper integrals in a more comprehensive manner.

• Includes Infinity: While infinity (\( +\infty\) and \( -\infty\) ) is not a real number, its inclusion allows us to handle limits that would otherwise be undefined or unbounded.
• Simplified Discussions: This makes it easier to discuss sequences and functions that tend towards very large or very small values without necessarily "blowing up" or becoming undefined.
• Extended Operations: The arithmetic and ordering of these numbers enable analyses of convergence and continuity when considering limits that involve infinity, such as \(\lim_{{n \to \infty}} x_n = +\infty\).

The functionality of the extended real numbers is further enhanced by metrics, which define how these numbers interact in terms of distances, as shown in metric space exercises.

Triangle Inequality

The triangle inequality is a fundamental property of a metric space. It serves as a tool to measure and relate the distances between three points in any given space.
The triangle inequality states that for any points \(x, y, z\), the following must hold:\[ d(x, z) \leq d(x, y) + d(y, z) \]This means that the direct distance between two points should always be the shortest. When taking a detour via a third point, the total distance should not be less than the direct path.

Importance in Real Analysis

• Ensures Consistency: This inequality assures that our distance notions remain consistent with geometric intuition.
• Useful in Proofs: It is a critical property often employed in proofs to show convergence, continuity, and other limits-based properties.
• Applications: Beyond pure mathematics, the triangle inequality has practical use in fields like computer science, especially in algorithms related to optimization and graph theory.

Understanding the triangle inequality within the context of metric spaces and extended real numbers helps in grasping the more abstract concepts of real analysis.