Fix a perfectoid field $K$ in mixed characteristic with ring of integers $\mathcal{O}$ and pseudo-uniformizer $\varpi$. Its tilt is the fraction field of $\mathcal{O}^{\flat}=\varprojlim_{x\mapsto x^{p}}\mathcal{O}/\varpi$. Pick an element $\pi\in\mathcal{O}^{\flat}$ such that $\pi^{\sharp}/p\in\mathcal{O}^{\times}$.

I want to understand Fontaine's infinitesimal period ring $$ A_{\text{inf}}:=W(\mathcal{O}^{\flat}). $$ Many references claim that it is complete with respect to the $(p,[\pi])$-adic topology. However, I was not able to find a reference for this statement by myself. I would be grateful if someone could provide a full proof here.