Hello.

I have the following function:

where is already a metric on some set and I am asked to prove that is also a metric on .

It is easy enough to show that is symmetric and positive definite, however, I am having problems with showing that satisfies the triangle identity.

I have tried a couple of different things (lots of algebra, transforming into a geo series, etc...) but either I keep on making mistakes or I am heading down the wrong path because I can never get it to reduce the way I want it too.

Any hints would be very much appreciated.

Thanks.