algebraic topology problem

Hi,

I'm taking first steps in algebraic topology and I became stuck at the problem:

"Prove that any map of the real projective space RP^n for n>=2 to S^1 is null homotopic" ("Algebraic topology" by E. Spanier. Ex.4 chapter2)

I've already proved that any two maps from a simply connected locally path-connected space to S^1 are homotopic (Ex.3 from same chapter).

Below I sketch what I have done. (I'm afraid this path isn't homotopic with a solution...). Pleas help :)!

S^n is (simply connected) double cover of RP^n so from the above we have that any two maps from S^n to S^1 are homotopic. In particular, for an arbitrary function g:S^n -> S^1 there exists a homotopy

H: S^nxI -> S^1 such that:

H(x,0)=g(x) for all x in S^n

H(x,1)=p for all x in S^n (p is some fixed point from S^1)

Now, for an arbitrary function f: RP^n -> S^1 I define a homotopy

F: RP^nxI -> S^1 by:

F(x,t)=H(y,t) where:

H is a homotopy given above where g is a composition of a covering projection c: S^n -> RP^n with function f. As c is a double cover I need to specify which point y choose to compute H(y,t).(for RP^2 it is quite easy, but what for n greater than 2?)

Looking forward for any clues and comments

magda