If $\alpha : [0,B) \rightarrow M, B \leq \infty$, is an extendible, piecewise smooth (nonspacelike curve) in a Lorentz manifold, then $\alpha$ has a finite length.
Any hints on how to show this? I know what all the words mean, but I have trouble seeing the trees. I tried writing out the arc length integral and playing around with the limits (using extendibillity), but it got me nowhere.
It is enough to treat the case of finite B, by reparametrizing.
But in this case, extending $\alpha$ past B, you can see that the length of $\alpha$ from $0$ to B is an integral of a continuous function ($\|\alpha'(t)\|$) over the compact set $[0, B]$, which is finite.