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Chapter 1: Problem 6

Show that the set of all real numbers constitutes an incomplete metric space if we choose $$ d(x, y)=|\arctan x-\arctan y| $$

Short Answer

Expert verified
The real numbers with \(d(x, y) = |\arctan(x) - \arctan(y)|\) form an incomplete metric space because not all Cauchy sequences converge within it.

Step by step solution

01

Understand the definition of a metric space

A metric space consists of a set alongside a metric (or distance function) that measures the distance between any two elements in the set. For a space to be a metric space, the metric must satisfy four properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.
02

Verify if d(x, y) is a valid metric

Given the metric \(d(x, y) = |\arctan(x) - \arctan(y)|\), we need to ensure it satisfies the metric properties. Non-negativity is satisfied as absolute values are non-negative. Identity of indiscernibles is verified: \(d(x, x) = 0\). Symmetry holds since absolute value is symmetric: \(d(x, y) = d(y, x)\). Lastly, check the triangle inequality: for any real numbers \(x, y, z\), \(d(x, z) \leq d(x, y) + d(y, z)\) holds true owing to the properties of the absolute value and the arctan function.
03

Identify completeness criteria

A metric space is complete if every Cauchy sequence converges to a limit within the space. A sequence \((x_n)\) is Cauchy if, for any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n > N\), \(d(x_n, x_m) < \epsilon\).
04

Test with a Cauchy sequence

Consider the sequence \(x_n = n\), i.e., the sequence of natural numbers. Under the usual arctan metric, \(d(x_n, x_m) = |\arctan(n) - \arctan(m)|\). As \(n, m \to \infty\), \(\arctan(n)\) and \(\arctan(m)\) both approach \(\frac{\pi}{2}\). Therefore, for any given \(\epsilon > 0\), we can choose \(N\) large enough such that \(|\arctan(n) - \arctan(m)| < \epsilon\) for all \(n, m > N\), confirming that \((x_n)\) is a Cauchy sequence.
05

Check convergence within the space

While \((x_n)\) is Cauchy, it does not converge within the set of real numbers with \(\arctan(x)\) as convergence implies a real limit, but \(\arctan(x_n)\) only approaches \(\frac{\pi}{2}\), not achievable within the real number set itself with real indices. Thus, the space is not complete, as not all Cauchy sequences converge within the space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Metric Space
A metric space provides a way of measuring distance between elements of a set, helping us establish structure numerically. It consists of a set paired with a distance function, known as a metric. This metric must satisfy four critical properties:
  • Non-negativity: The distance between any two points must always be zero or positive. Mathematically, for any points \(x\) and \(y\), this is expressed as \(d(x, y) \geq 0\).
  • Identity of indiscernibles: This implies that a distance of zero means no difference; i.e., \(d(x, y) = 0\) if and only if \(x = y\).
  • Symmetry: The metric should work both ways equally, so \(d(x, y) = d(y, x)\).
  • Triangle inequality: The shortest distance between two points is a straight line, which autotranslates to \(d(x, z) \leq d(x, y) + d(y, z)\).
In the context of our exercise, these properties are tested using the metric \(d(x, y) = |\arctan(x) - \arctan(y)|\). Each property was satisfied, verifying it's a valid metric.
Cauchy Sequence
A Cauchy sequence is an essential concept in analysis when exploring completeness. Such a sequence ensures that its elements are getting arbitrarily close to each other as they progress.For a sequence \((x_n)\) to be considered Cauchy, the following must be true:
  • For every positive number \(\epsilon\), no matter how small, there exists a number \(N\) such that for any \(m, n > N\), the distance \(d(x_n, x_m) < \epsilon\).
In simpler terms, once you travel far enough down the sequence鈥攂eyond some threshold鈥攁ll subsequent terms will be as close to each other as desired.This concept was applied to the sequence of natural numbers using the given arctan metric, demonstrating that even as numbers grew larger, their arctan values became arbitrarily close.
Arctan Function
The arctan function, known as the inverse tangent, provides the angle whose tangent is a given number. It is particularly intriguing because, unlike linear functions, it approaches asymptotic limits rather than extending linearly.Important characteristics of the arctan function include:
  • Its values range between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
  • As the input values approach both positive and negative infinity, the function approaches but never quite reaches these asymptotic limits.
  • This behavior gives rise to interesting properties when used within metrics, particularly in regard to convergence.
In the solution, recognizing how arctan values approach \(\frac{\pi}{2}\) as inputs grow illustrates why sequences like natural numbers appear to converge under this metric.
Completeness Criteria
In mathematics, especially in analysis, completeness is a critical property of metric spaces, determining whether every Cauchy sequence not only converges but converges to a limit that exists within the space. To assess completeness, we pursue the following steps:
  • Identify or create a Cauchy sequence within the space.
  • Determine if the sequence converges to a limit within the same space.
If the sequence converges within the space, it confirms completeness; otherwise, incompleteness is indicated. Regarding the exercise, while a sequence formed from natural numbers was Cauchy in the set of real numbers under the arctan metric, it didn't converge to a limit within the real numbers themselves, showcasing the space's incompleteness.

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Most popular questions from this chapter

In 1.5-5 we referred to the following theorem of calculus. If a sequence \(\left(x_{m}\right)\) of continuous functions on \([a, b]\) converges on \([a, b]\) and the convergence is uniform on \([a, b]\), then the limit function \(x\) is continuous on \([a, b]\). Prove this theorem.

A homeomorphism is a continuous bijective mapping \(T: X \longrightarrow Y\) whose inverse is continuous; the metric spaces \(X\) and \(Y\) are then said to be homeomorphic. (a) Show that if \(X\) and \(Y\) are isometric, they are homeomorphic. (b) Illustrate with an example that a complete and an incomplete metric space may be homeomorphic.

Show that nonnegativity of a metric follows from (M2) to (M4).

Does \(d(x, y)=(x-y)^{2}\) define a metric on the set of all real numbers?

Show that a Cauchy sequence is bounded.
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