In the article "What is ... Equivariant Cohomology?" by Loring W. Tu (https://arxiv.org/abs/1305.4293), I read that
$H^*(\mathbb{C}P^{\infty},\mathbb{R})\simeq \mathbb{R}[u]$,
where $H^*()$ denotes the singular cohomology (functor), $\mathbb{C}P^{\infty}=\cup_n \mathbb{C}P^{n}$ and $\mathbb{R}[u]$ is the polynomial ring generated by an element $u$ of degree 2.
I don't see this equivalence. Can you explain me why it is so?