Why is this pseudometric well defined? Let $M\subset\Bbb C^n$ be a bounded domain. Let $\Bbb D$ be the open unit disc in the complex plane. Let $ H(\Bbb D, M)$ denote all the holomorphic maps from $\mathbb{D}$ to $M$ . Let $w\in M$ and $d\in \Bbb C^n$ . The Kobayashi-Royden psedometric, denoted by $k_M$ is defined as $$k_M(w,d)= \inf \{ l\in(0,\infty) \; | \; \exists \phi\in H(\Bbb D, M): \phi( 0)=w, l\phi’(0)=d\}. $$
Can any one tell me why this function is well defined?  As to why the set over which the infimum is taken non empty?
The proof of the Rieman mapping Theorem can be used.

The set is not empty because for $C$ small enough the map
$$z\mapsto w+  d\cdot (Cz)$$
is a holomorphic map $\Bbb D\to M$ with $\phi(0)=w$ and $\frac1C \phi'(0) = d$, hence $\frac1C$ is in the set over which you take the infimum. The only problem is ensuring that the image of this map is contained in $M$, this can be achieved by making $C$ small enough because $M$ is open. An example of a $C$ that is small enough is given by $C=\frac1{\|d\|}\mathrm{dist}(w, \partial M)$.
