5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1, a2,..., an € M.
5. In any metric space (M, D), prove that D(a1, an) ≤ D(a₁, a2) + D(a2, a3) + + D(an-1, an), for a1, a2,..., an € M.
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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Question
Solve number 5
Transcribed Image Text:5. In any metric space (M, D), prove that D(a₁, an) ≤ D(a1, a2) +
D(a2, a3) + + D(an-1, an), for a1, a2,..., an € M.
6. A metric space (M, D) is called ultra-mteric if D(a, c) max
{D(a, b), D(b, c)} for all a, b, c, M. Prove that the following met-
ric is ultra-metric: M is any set and D(a, a) = 0 and D(a, b):
= 1 for
a bin M.
Expert Solution
Step 1
Given is a metric space.
The triangle inequality holds in a metric space, which means for
we have .
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