# Any embedding $\mathbb{R} P^n\hookrightarrow \mathbb{R} P^{\infty}$ induces a surjection of the singular cohomology rings.

On page 79 of "Vector Bundles and K-Theory" Hatcher makes use of the claim in the title where he considers cohomology with integer or $$\mathbb{F}_2$$ coefficients.

Why is this claim true?

I know that it suffices to see that it induces a surjection on the first cohomology module.

This is not true for arbitrary embeddings. It is true for embeddings that come from a linear map $$\mathbb{R}^n\to\mathbb{R}^\infty$$. First, if $$i:\mathbb{R}^n\to\mathbb{R}^\infty$$ is the standard inclusion then the induced inclusion $$\mathbb{R}P^n\to\mathbb{R}P^\infty$$ is just the inclusion of the $$n$$-skeleton for the usual CW-complex structure on $$\mathbb{R}P^\infty$$ and so the conclusion follows easily using cellular cohomology. Now if $$j:\mathbb{R}^n\to\mathbb{R}^\infty$$ is any other linear injection, there is a linear automorphism $$T:\mathbb{R}^\infty\to\mathbb{R}^\infty$$ such that $$i=Tj$$. Since $$T$$ induces an homeomorphism from $$\mathbb{R}P^\infty$$ to itself, the embedding $$\mathbb{R}P^n\to\mathbb{R}P^\infty$$ induced by $$j$$ differs from the one induced by $$i$$ by composition with a homeomorphism, and so also induces surjections on mod $$2$$ cohomology.
(Alternatively, as Hatcher mentions, you can show that $$i$$ and $$j$$ are homotopic through linear injections, which induces a homotopy between the inclusions $$\mathbb{R}P^n\to\mathbb{R}P^\infty$$. Namely, if $$k:\mathbb{R}^n\to\mathbb{R}^\infty$$ is a linear injection whose image has trivial intersection with the images of $$i$$ and $$j$$, then you can take the straight-line homotopies from $$k$$ to each of $$i$$ and $$j$$.)