Problem 11 - Basic Analysis: Introduction to Real Analysis

Question

a) Find a metric d on \(\mathbb{N}\), such that \(\mathbb{N}\) is an unbounded set in \((\mathbb{N}, d)\). b) Find a metric \(d\) on \(\mathbb{N}\), such that \(\mathbb{N}\) is a bounded set in \((\mathbb{N}, d)\). c) Find a metric \(d\) on \(\mathbb{N}\) such that for any \(n \in \mathbb{N}\) and any \(\varepsilon>0\) there exists an \(m \in \mathbb{N}\) such that \(d(n, m)<\varepsilon\).

Step-by-Step Solution

Answer

a) Metric: \(d(m, n) = |m-n|\) is unbounded. b) Metric: Discrete metric is bounded. c) Metric: \(d(m, n) = \frac{1}{1+|m-n|}\) satisfies the \(\varepsilon\)-condition.

Step 1: Understand the Concepts

In this exercise, we're dealing with the set of natural numbers \(\mathbb{N}\) and different types of metric spaces. A metric \(d\) on a set \(X\) is a function \(d: X \times X \to \mathbb{R}\) satisfying certain conditions: non-negativity, identity of indiscernibles, symmetry, and triangle inequality. We need to find such metrics with specific properties based on these conditions.

Step 2: Part (a) Find an Unbounded Metric

Let's consider the standard Euclidean metric on \(\mathbb{N}\), defined as \(d(m, n) = |m - n|\). Because the natural numbers are infinite and differ increasingly as their value grows, \((\mathbb{N}, d)\) is an unbounded set under this metric, as there is no finite number that can bound the differences \(|m - n|\) for all \(m, n \in \mathbb{N}\).

Step 3: Part (b) Find a Bounded Metric

A simple way to define a bounded metric is to use a discrete metric: \(d(m, n) = 1\) if \(m eq n\) and \(d(m, n) = 0\) if \(m = n\). In this case, \(\mathbb{N}\) is bounded because the maximum distance between any pair of elements is 1.

Step 4: Part (c) Construct the Metric Satisfying Limit Condition

To satisfy this condition, use a metric that compresses distances, such as \(d(m, n) = \frac{1}{1 + |m - n|}\). Here, for any \(n \in \mathbb{N}\) and any \(\varepsilon > 0\), you can always find a sufficiently large \(m\) such that \(|m-n|\) is large enough for \(d(n, m) < \varepsilon\). As \(|m-n|\) grows, \(\frac{1}{1+|m-n|}\) becomes arbitrarily small.

Key Concepts

Natural Numbers

Natural numbers, often denoted by the symbol \(\mathbb{N}\), form the most basic set of numbers used in mathematics. This set includes the positive integers starting from 1, 2, 3, and so on. Natural numbers are

• The building blocks of arithmetic operations.
• Used for counting and ordering.
• An infinite set, meaning they continue indefinitely without bound.

When discussing sets like the natural numbers in the context of metrics, it's important to understand their infinite nature. This infinity plays a crucial role when considering metrics that make a set bounded or unbounded. By default, without any specific constraints, the natural numbers can stretch out infinitely.

Euclidean Metric

The Euclidean metric is one of the most commonly used metrics in mathematics, especially in the context of real number spaces and natural numbers. Defined as \(d(m, n) = |m - n|\), this metric measures the "straight-line" distance between two points \(m\) and \(n\) in any given space, including the set of natural numbers.

• It obeys basic metric properties: non-negativity, identity of indiscernibles, symmetry, and triangle inequality.
• Often used in calculus and geometry to calculate distances.

When applied to natural numbers, the Euclidean metric identifies how far apart two numbers are in terms of integer steps. The infinite nature of natural numbers makes them unbounded under the Euclidean metric since differences between numbers can grow indefinitely.

Discrete Metric

The discrete metric offers a fascinating contrast to the Euclidean metric. Defined simply, it assigns a distance of 1 if two numbers \(m\) and \(n\) are different, and 0 if they are the same, mathematically represented as:

• \(d(m, n) = 1\) if \(m eq n\)
• \(d(m, n) = 0\) if \(m = n\)

This metric is particularly interesting because:

• Every pair of distinct points in the space is equidistant, simplifying analysis.
• The entire set becomes bounded since the maximum possible distance between any two points is 1.

For the set of natural numbers, the discrete metric provides an easy way to represent all differences between distinct elements as uniformly minimal. This is essential in exercises aiming to bound the set \(\mathbb{N}\).

Unbounded Set

An unbounded set in the context of metric spaces refers to a set where there is no finite limit to the distances between its elements. In simpler terms, elements within an unbounded set can be arbitrarily far apart from each other as you move along the set.

• In our context, natural numbers become unbounded with the Euclidean metric.
• This occurs because the distance \(|m - n|\) can increase indefinitely as you choose larger and larger \(m\) and \(n\).

Unboundedness is an intrinsic property of the metrics that do not impose an absolute cap on how large distances can become. For sets like \(\mathbb{N}\), using the Euclidean metric, it highlights the infinite expansiveness of natural numbers.

Bounded Set

In contrast to unbounded sets, a bounded set in metric spaces has a maximum finite distance among its elements. This means there exists a real number \(M\) such that no pair of elements in the set is separated by a distance exceeding \(M\).

• The discrete metric can make natural numbers bounded.
• Under this metric, the maximum distance between any two natural numbers is always 1.

Bounding a set like \(\mathbb{N}\) transforms its infinitely detailed structure into a more controlled structure where differences are predictable and constant. Such bounded properties are crucial in various mathematical disciplines, allowing us to constrain calculations and reasoning within a finite scope.