Approximation of Teichmuller lifts by iterated pth powers ($v_K(y_n^{p^n}-[x])\ge n+1$)

For $K/{\mathbb{Q}}_p$ finite, let $x_n^{p^n}=x$ in its finite residue field, and choose arbitrary lifts $y_n\in {\mathcal{O}}_K$. With normalized discrete valuation,Applying this binomial estimate repeatedly to $y_n$ and $[x_n]$ proves the displayed bound, and hence convergence to $[x]$.