I want to show that the unit disk equipped with the hyperbolic metric is a complete metric space.
The metric is defined as:
$d_{P}(w,z)=inf\{l_{h}(\gamma): \gamma$ is a partial smooth curve in $\mathbb{D}$ with initial point at z and ending point w$\}.$
I want to show that every Cauchy sequence is convergent but I find it difficult.
I also know that the hyperbolic metric is equivalent to the Euclidean metric but I do not think that this is useful since completeness is not a topological property.
Any answer or discussion on that would be really useful.