Real Analysis We started metric spaces yesterday. I am asked to prove that the metric space defined as P and Q is a metric space. When I look at examples of these proofs they all say that the proof of...

Real Analysis

We started metric spaces yesterday.  I am asked to prove that the metric space defined as P<>₁,y₁> and Q<>₂,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<>₃,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?