There are some details I still don't understand about the definition of Stiefel - Whitney / Chern - classes. Let $\gamma_n^1$ be the tautological line bundle over the real or complex projective space $\mathbb{P}^n$. For $n\to\infty$ write $\gamma^1$ instead of $\gamma_\infty^1$. There are two common ways to normalize Stiefel - Whintey classes, either require $$w_1(\gamma_1^1)\neq 0\text{ (e.g. Milnor, Stasheff)}\quad\text{or that}\quad w_1(\gamma^1) \text{ generates } H^1(\mathbb{R}P^\infty;\mathbb{Z}_2)\text{ (e.g. Hatcher)}$$
Why are these definitions equivalent? Say you use Milnor/Stasheff's definition, then the inclusion $j:\mathbb{R}P^1\hookrightarrow\mathbb{R}P^\infty$ is covered by a bundle map such that $\gamma_1^1 \cong j^*\gamma_1$ and hence $j^*w_1(\gamma_1) = w_1(\gamma_1^1)\neq 0$ requires $w_1(\gamma_1)\neq 0$ (right?)
Vice versa, Hatcher writes (p.81, Vector bundles and K - theory) "...the inclusion $i:\mathbb{R}P^1\hookrightarrow \mathbb{R}P^\infty$ induces an isomorphism $H^1(\mathbb{R}P^\infty;\mathbb{Z}_2) \cong H^1(\mathbb{R}P^1;\mathbb{Z}_2)$" and I can't see why this is true.
There is a more general statement that $H^p(G_n(\mathbb{R}^{n+k})) \cong H^p(G_n(\mathbb{R}^\infty))$ for all $p<k$ and I know why this is true ( (co)homology of CW - complexes) but it doesn't apply to Hatcher's statement since then $n=k=p=1$.
I appologize if this is something trivial, I am still a beginner with these things. Thanks for any help!