First of all, X should be compact. Note that is closed in X. Also, as a is a regular value, F must be one to one over a neighbourhood of each element in .
In a more involved proof, I am getting caught up on this single step.
Let f: X --> R^(2k) be an immersion and F: T(X) --> R^(2k), where T(X) is the tangent bundle, and F(x,v) = df_x (v) (it is the differential of F at x on the vector v.) Let "a" be a regular value of F. Show that F inverse of a is a finite set.
Any suggestions on how to show it is finite? Thanks a lot.