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?