Given a metric space $(X,d)$ the following is a definition of a geodesic from the book of Santambrogio.
I want to show that a constant speed geodesic is a geodesic : is the following enough?
Note that \begin{align*} \text{Length}(\omega)=\sup_{\text{partitions}} \sum_{i=1}^n d\big(\omega(s_{i-1}),\omega(s_i)\big) = \sup_{\text{partitions}} d\big(\omega(t_{0}),\omega(t_{1})\big) \sum_{i=1}^n \frac{t_1-t_{0}}{s_{i}-s_{i-1}}=d\big(\omega(t_{0}),\omega(t_{1})\big), \end{align*} since we are in a length space this curve achieves the infimum, and hence must be a geodesic.