ScalingStacks

Proof. If $\mathcal{P}\not\simeq\mathbb{E}_{0}$, then, by 3.2.3, $\mathcal{P}$ is $d$-connected for some $d\geq-1$. Therefore, by 5.2.2, $\mathcal{P}\otimes\mathcal{P}$ is $\left(2d+2\right)>d$ connected. Since $\mathcal{P}\simeq\mathcal{P}\otimes\mathcal{P}$, we can continue by induction and deduce that $\mathcal{P}$ is $\infty$-connected; hence $\mathcal{P}\simeq\mathbb{E}_{\infty}$. ∎