This might be a very trivial question so bare with me. If $(X,d)$ is a length space we define a unit speed geodesic to be a path $\gamma:[0,1]\to X$ for which \begin{align*} d(\gamma(s),\gamma(t))=|t-s|d(\gamma(0),\gamma(1))\,\,\mathrm{for}\,\,\mathrm{all}\,\,s,t\in[0,1]. \end{align*} The author notes without further remarks that this is equivalent to if $\leq$ always a holds. I tried couple of tricks and I would like some fresh input if possible. I'm literally stuck and I believe there's a very simple way to see why this holds.
I know that we can find a reparameterization $\widetilde{\gamma}$ of $\gamma$ (i.e. $\gamma=\widetilde{\gamma}\circ\alpha$ for some non-decreasing continuous surjection $\alpha:[0,1]\to[0,1]$) for which the equation holds. From here to me it seems we can't do better than $d(\gamma(s),\gamma(t))=|\alpha(t)-\alpha(s)|d(\gamma(0),\gamma(1))$ for all $s,t\in[0,1]$. Or can we conclude something from here?
If someone is interested on the source, it's "A user's guide to optimal transportation", written by Ambrosio and Gigli. It's probably free to view through google. Page 34, equation (2.5).
Thanks in advance.