Map from Sphere to RP^2 contractible?

Question

Since there are some experienced math guys about, here's a fun question: Prove that every map f: S^2--->RP^2 is contractible (without just quoting a theorem).

Oops! I meant to say every map from RP^2--->S^2. Sorry about that.

Again, I meant to say that all maps from RP^2 to S^2 are contractible, where RP^2 is indeed the real projective plane.

Here's a hint: Look at what happens when you pull back the Euler class of the tautological bundle on S^2 to RP^2.

Answer 1

I am assuming that you are talking about the real projective plane. So I am guessing that you use a quotient map or a projection.