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Consider a set A and a function d: A × A → R that satisfies: • d(x, y) = 0 ⇔ x = y; • d(x, y) = d(y, x); • d(x, y) ≤ d(x, z) + d(z, y). Prove that (A, d) is a metric space, i.e. show that d(x, y) ≥ 0 for all x, y ∈ A.

Consider a set A and a function d: A × A → R that satisfies: • d(x, y) = 0 ⇔ x = y; • d(x, y) = d(y, x); • d(x, y) ≤ d(x, z) + d(z, y). Prove that (A, d) is a metric space, i.e. show that d(x, y) ≥ 0 for all x, y ∈ A.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 24CM
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Consider a set A and a function d: A × A → R that satisfies:
• d(x, y) = 0 ⇔ x = y;
• d(x, y) = d(y, x);
• d(x, y) ≤ d(x, z) + d(z, y).
Prove that (A, d) is a metric space, i.e. show that d(x, y) ≥ 0 for
all x, y ∈ A.

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