Comformal map in hyperbolic space $\mathbb{H}^n$, $n\geq 3$.

Question

I'm working with some result in Do carmo, Riemannian Geometry. Which consits in to find all isometries in $\mathbb{H}^n$, but in the case $\mathbb{H}^n$, $n \geq 3$, the autor said the comformal map $h$ maps $\partial \mathbb{H}^n$ to $\partial \mathbb{H}^n$. In a first place I think that statement was due some special property of comformal maps. But I find this

Does a conformal map take boundaries to boundaries?

So, I don't know how prove this affirmation. Some hint?

Thanks in advance!