Limits of Amao multiplicities and Epsilon multiplicity

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and $I$ be an ideal of $R$. The epsilon multiplicity $\epsilon (I)$ of $I$ is shown in a recent paper of Stephen Landsittel to be the limit of Amao multiplicities $$\epsilon (I)=\underset{m\to \mathrm{\infty }}{\lim }\frac{a(I^m,(I^m{)}^{\mathrm{s}\mathrm{a}\mathrm{t}})}{m^d}$$ if the dimension of the nilradical of $\widehat{R}$ is less than $d$. It is shown in this paper that for an arbitrary local ring $R$ which is of finite type over a field, the limit $$\underset{m\to \mathrm{\infty }}{\lim }\frac{a(I^m,(I^m{)}^{\mathrm{s}\mathrm{a}\mathrm{t}})}{m^d}$$ always exists.