Verify that d : given by d(X,Y) := for defines a distance on
So, I haven't actually learned this yet in class... so I don't really understand what this is saying...
Originally Posted by Plato
You must show three properties.
The first two are trivial. The third may be a bit tricky to prove if you get hung up on the absolute value signs. All you need to know is that is a metric. It would be easiest to label them: , , . You know from the triangle inequality that .
So now here's what you have to prove:
given that .
Make common denominators and combine the fractions: .
You only have to show that the numerator is positive, as the denominator is clearly positive (because ).
Multiply out the numerator:
Now we use the fact that to say that .
Clearly , because , , and are all positive. So we're done.