Hello.

I have the following function:

$\displaystyle d_1(x,y)=d(x,y)/(1+d(x,y))$

where $\displaystyle d(x,y)$ is already a metric on some set $\displaystyle X$ and I am asked to prove that $\displaystyle d_1(x,y)$ is also a metric on $\displaystyle X$.

It is easy enough to show that $\displaystyle d_1(x,y)$ is symmetric and positive definite, however, I am having problems with showing that $\displaystyle d_1(x,y)$ 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.