Gysin proof of mod-two projective-space cohomology ($H^{\ast }({\mathbb{RP}}^n;{\mathbb{F}}_2)={\mathbb{F}}_2[a]/(a^{n+1})$)

The unit sphere bundle of the tautological line is the double cover $S^n\to {\mathbb{RP}}^n$. In its Gysin sequence, multiplication by the mod-two Euler class $a$ is an isomorphism in successive interior degrees, since the sphere has no interior cohomology. Connectedness in degree zero makes multiplication into degree one injective. At the top, vanishing above dimension $n$ makes the transfer from $H^n(S^n;{\mathbb{F}}_2)$ surjective; together with that injection it makes the top group one-dimensional. Thus every $a^j$, $0\le j\le n$, is the nonzero generator and the next power vanishes.