Real Analysis We started metric spaces yesterday. I am asked to prove that the metric space defined as P<x1,y1> and Q<x2,y2> is a metric space. When I look at examples of these proofs they all say that the proof of the first properties (positive definiteness and symmetry) are trivial. I think I must be more complete than that. How do I best show that these properties are true for this space? Further, how do I show that the triangle inequality is true? Do I create a new ordered pair and call it R<x3,y3> and then prove that the distance between P and Q is less than or equal to the distances from P to R + R to Q?
Real Analysis We started metric spaces yesterday. I am asked to prove that the metric space defined as P<x1,y1> and Q<x2,y2> is a metric space. When I look at examples of these proofs they all say that the proof of the first properties (positive definiteness and symmetry) are trivial. I think I must be more complete than that. How do I best show that these properties are true for this space? Further, how do I show that the triangle inequality is true? Do I create a new ordered pair and call it R<x3,y3> and then prove that the distance between P and Q is less than or equal to the distances from P to R + R to Q?
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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We started metric spaces yesterday. I am asked to prove that the metric space defined as P<x1,y1> and Q<x2,y2> is a metric space. When I look at examples of these proofs they all say that the proof of the first properties (positive definiteness and symmetry) are trivial. I think I must be more complete than that. How do I best show that these properties are true for this space?
Further, how do I show that the triangle inequality is true? Do I create a new ordered pair and call it R<x3,y3> and then prove that the distance between P and Q is less than or equal to the distances from P to R + R to Q?
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