Let $\pi: \mathbb{S}^{2n+1} \subset \mathbb{C}^{n+1} \to \mathbb{CP}^n$ be given by $\pi(z_0, \cdots, z_n) = [z_0, \cdots, z_n]$ ($\pi$ is called the Hopf fibration). Prove that $\pi$ admits no global sections, that is, there does not exist any $s: \mathbb{CP}^n \to \mathbb{S}^{2n+1}$ such that $\pi \circ s = \operatorname{Id}_{\mathbb{CP}^n}$.
I'm aware a similar question has been asked before, but that question does not solve mine and it's only similar, not the exact same. Now, here's how I went about solving this (I want to know if it's all correct): if there were such a section, then the composition $$\mathbb{R} = H^{2n}(\mathbb{CP}^n) \stackrel{\pi^{*}}{\to}H^{2n}(\mathbb{S}^{2n+1}) = 0\stackrel{s^{*}}{\to} H^{2n}(\mathbb{CP}^n) = \mathbb{R}$$
is the identity on $H^{2n}(\mathbb{CP}^n) $, which is evidently a contradiction since there's a $0$ in the middle. Here the cohomologies I'm working with are all de-Rham cohomology.