# Thread: Help with Metric Spaces?

1. ## Help with Metric Spaces?

I have a two part question that I'm not quite sure how to approach... it goes as follows:

1. Prove that (R^n, d) is a metric space, where:
d_mb(x,y) =d_m(x,y)/(d_m(x,y) +1)

2. Find the unit ball B_M(0; 1) for M = (R^3, d), where d is the distance function defined in 1.

Any help would be appreciated! Thank you!

2. Assuming $d_m$ is a metric $d_{mb}:=d$ is also a metric since $d(x,y)=0$ iff $d_m(x,y)=0$ iff $x=y$ and clearly $d \geq 0$. The commutativity follows immediately and so we only have to prove the triangle inequality which is equivalent to: $d_m(x,z)(d_m(x,y)+1)(d_m(y,z)+1) \leq (d_m(x,z)+1)(d_m(x,y)+1)$ which is clearly true.

For the second one, which is $d_m$? Is it the usual metric?

3. Originally Posted by Majialin
I have a two part question that I'm not quite sure how to approach... it goes as follows:

1. Prove that (R^n, d) is a metric space, where:
d_mb(x,y) =d_m(x,y)/(d_m(x,y) +1)

2. Find the unit ball B_M(0; 1) for M = (R^3, d), where d is the distance function defined in 1.

Any help would be appreciated! Thank you!
2. It is all of $\mathbb{R}^n$, because $\forall~x,y\in\mathbb{R}^n$, $d_{mb}(x,y)<1$.