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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.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.2: Ring Homomorphisms
Problem 6E
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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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