2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 + d(x, y) for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and that ρ(x, y) ≤ d(x, y) for all x, y ∈ S.
2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by ρ(x, y) = d(x, y) 1 + d(x, y) for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and that ρ(x, y) ≤ d(x, y) for all x, y ∈ S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined by
ρ(x, y) = d(x, y)
1 + d(x, y)
for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and that
ρ(x, y) ≤ d(x, y) for all x, y ∈ S.
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