I have a question about an exercise in [Misner, Thorne, and Wheeler's *Gravitation*](https://en.wikipedia.org/wiki/Gravitation_(book)). On page 321, exercise 13.5 says

> 
> Show that a geodesic of spacetime which is timelike at one event is
>  everywhere timelike. Similarly, show that a geodesic initially
>  spacelike is everywhere spacelike, and a geodesic initially null is
>  everywhere null. [*Hint:* This is the easiest exercise in the book!]
> 
> 
> 

Ok so the idea is that timelike means that the timelike geodesic have a negative-length 4-velocity, a spacelike would have a positive-length 4-velocity and a null would have a 0-length 4-velocity. So my idea was to use the definition of the geodesic to show that the sign of the length of the 4-velocity doesn't change. The definition of the geodesic translates to a curve that parallel transport conserves the tangent vector to the curve.

My idea was to say that the tangent vector of a spacelike is always in the space parts of spacetime. But it doesn't seem true at all. Finally the Hint doesn't help me much.

I assume there are two kinds of solutions for this problem: one purely abstract with words and one with a more mathematical approach of the concept of geodesics and tangent vectors.