Immersion and Manifold Question

[evalover1987]

Homework Statement

Let's assume that M is a compact n-dimensional manifold,
then from Whitney's Immersion Theorem,
we know that there's an immersion, f: M -> R_2n, and
let's define f*: TM --> R_2n such that
f* sends (p, v) to df_x (v).
Since f is an immersion, it's clear that f* must be one-to-one by definition of immersion.
let "x" be a regular value of f*, then how would you show that
the inverse image of x (with respect to f, not f*) consists of finitely many points?

Homework Equations

The Attempt at a Solution

I reduced the problem to this. Once I show that there are only finitely many preimages (with respect to f, not f*) of
x in the compact set C = {(p, v) in TM : |v| <= 1}, I'm done.
I tried to prove it using proof by contradiction, so
I assumed that there are infinitely many points in that set, then we obviously,
there's a subsequence such that
pi --> p vi --> w for some (p, w) in C
by properties of a compact set.
Then once I show that df_p(w) = 0, I'm done, but
I'm struggling to show how that would work.

 

[kurt.physics]

I read somewhere that df_p(w) = 0 iff w is tangent to f(M), but I'm having trouble understanding the implications of that statement.Any help would be greatly appreciated, thank you in advance!