I have seen the usual definition of the distance in a Riemannian manifold $M$, considering piecewise $C^\infty$ curves.
I know that we consider this kind of curves only for convenience (for example it is a device to have a simpler proof of the triangle inequality).
If I consider the metric $d$ defined using $C^\infty$ curves, how can I prove the tringle inequality? How can I round off corners of a piecewise path so that the length increases less than any given $\epsilon$?
The only idea I had is using mollificators but I don't know if I can use them here...Thanks!