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Exercise 1.4.8. The following construction generalizes the construction of the reals from the rationals in Chapter 5, allowing one to view any metric space as a subspace of a complete metric space. In what follows we let (X,d) be a metric space. n→ ∞ Yn (a) Given any Cauchy sequence (xn)x1 in X, we introduce the formal limit LIMn on. We say that two formal limits LIM∞ xn and LIMn- are equal if lim→∞ d(xn, Yn) is equal to zero. Show that this equality relation obeys the reflexive, symmetry, and transitive axioms. (b) Let X be the space of all formal limits of Cauchy sequences in X, with the above equality relation. Define a metric dx: XxX → R+ by setting dx (LIMn→∞ xn, LIM→∞ Yn) := lim_d(xn, Yn). n→X Show that this function is well-defined (this means not only that the limit lim→∞ d(xn, Yn) exists, but also that the axiom of substitution is obeyed; cf. Lemma 5.3.7), and gives X the structure of a metric space. (c) Show that the metric space (X, dx) is complete.

Exercise 1.4.8. The following construction generalizes the construction of the reals from the rationals in Chapter 5, allowing one to view any metric space as a subspace of a complete metric space. In what follows we let (X,d) be a metric space. n→ ∞ Yn (a) Given any Cauchy sequence (xn)x1 in X, we introduce the formal limit LIMn on. We say that two formal limits LIM∞ xn and LIMn- are equal if lim→∞ d(xn, Yn) is equal to zero. Show that this equality relation obeys the reflexive, symmetry, and transitive axioms. (b) Let X be the space of all formal limits of Cauchy sequences in X, with the above equality relation. Define a metric dx: XxX → R+ by setting dx (LIMn→∞ xn, LIM→∞ Yn) := lim_d(xn, Yn). n→X Show that this function is well-defined (this means not only that the limit lim→∞ d(xn, Yn) exists, but also that the axiom of substitution is obeyed; cf. Lemma 5.3.7), and gives X the structure of a metric space. (c) Show that the metric space (X, dx) is complete.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 24EQ
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Exercise 1.4.8. The following construction generalizes the construction of the reals from the rationals in Chapter 5, allowing one to view any metric space as a subspace of a complete metric space. In what follows we let (X,d) be a metric space. n→ ∞ Yn (a) Given any Cauchy sequence (xn)x1 in X, we introduce the formal limit LIMn on. We say that two formal limits LIM∞ xn and LIMn- are equal if lim→∞ d(xn, Yn) is equal to zero. Show that this equality relation obeys the reflexive, symmetry, and transitive axioms. (b) Let X be the space of all formal limits of Cauchy sequences in X, with the above equality relation. Define a metric dx: XxX → R+ by setting dx (LIMn→∞ xn, LIM→∞ Yn) := lim_d(xn, Yn). n→X Show that this function is well-defined (this means not only that the limit lim→∞ d(xn, Yn) exists, but also that the axiom of substitution is obeyed; cf. Lemma 5.3.7), and gives X the structure of a metric space. (c) Show that the metric space (X, dx) is complete.
Transcribed Image Text:Exercise 1.4.8. The following construction generalizes the construction of the reals from the rationals in Chapter 5, allowing one to view any metric space as a subspace of a complete metric space. In what follows we let (X,d) be a metric space. n→ ∞ Yn (a) Given any Cauchy sequence (xn)x1 in X, we introduce the formal limit LIMn on. We say that two formal limits LIM∞ xn and LIMn- are equal if lim→∞ d(xn, Yn) is equal to zero. Show that this equality relation obeys the reflexive, symmetry, and transitive axioms. (b) Let X be the space of all formal limits of Cauchy sequences in X, with the above equality relation. Define a metric dx: XxX → R+ by setting dx (LIMn→∞ xn, LIM→∞ Yn) := lim_d(xn, Yn). n→X Show that this function is well-defined (this means not only that the limit lim→∞ d(xn, Yn) exists, but also that the axiom of substitution is obeyed; cf. Lemma 5.3.7), and gives X the structure of a metric space. (c) Show that the metric space (X, dx) is complete.
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