Question

Asked Nov 23, 2019

Real Analysis

We started metric spaces yesterday. I am asked to prove that the metric space defined as P<x_{1},y_{1}> and Q<x_{2},y_{2}> 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<x_{3},y_{3}> 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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