I tried to understand this paper on page 31.

Let $K$ be an finite extension of $\mathbb Q_p$ and $\overline{K}$ be its algebraic closure; $\mathcal{O}_{\overline{K}}$ is the ring of integers of $\overline{K}$; $\mathcal{O}_{\mathbb{C}_K}$ is its $p$-adic completion. We have $R:=\mathcal{O}_{\overline{K}}/(p)\cong\mathcal{O}_{\mathbb{C}_K}/(p)$ canonically.

There is a natural projection onto the first component: $$\phi:R^{\mathrm{perf}}\to R,$$ where the $R^{\mathrm{perf}}$ contain elements of the form $(x_1,x_2,...)$ with $x_i^p=x_{i-1}$.

Now, use the universal propety of Witt vector we get a lift

$$\theta:W(R^{\mathrm{perf}})\to\mathcal{O}_{\mathbb{C}_K}.$$ And the ring $W(R^{\mathrm{perf}})$ is denoted by $A_{\mathrm{inf}}$.

I have two question about it:

Then the author claims $R^{\mathrm{perf}}$ is $\ker(\phi)$-adically complete, but I doubt whether it is true. If we consider the map. In fact, if we consider the system $\phi_n:R^{\mathrm{perf}}\to R$ by projecting onto the $n$-the component. It is easy to see $R^{\mathrm{perf}}$ is isomorphic to the completion this system $\{\ker(\phi_n)\}$. But I doubt $\ker(\phi_n)=\ker(\phi)^{n}$.

Even this is true how do I get the $\ker(\theta)$-adic completeness of $A_{\mathrm{inf}}$?