Observe that we have $\tau({\mathfrak{a}}^{\lambda})=({\mathfrak{a}}^{\lceil\lambda p^{e}\rceil})^{[1/p^{e}]}$ for large $e\in\mathbb{N}$ as $X$ is noetherian. The equality

(4.4)

$$\tau(({\mathfrak{a}}^{m})^{\lambda})=\tau({\mathfrak{a}}^{\lambda m})$$

for $m\in\mathbb{N}$ shows that the notation in Definition 4.1 is compatible with taking powers of ideals [BMS08, Cor. 2.15]. We have $\tau({\mathfrak{a}}^{\lambda})\subseteq\tau({\mathfrak{b}}^{\lambda})$ for ideals ${\mathfrak{a}}\subseteq{\mathfrak{b}}$ in ${\mathcal{O}}_{X}$ [BMS08, Prop. 2.11(i)].

Choose $e$ such that $\tau({\mathfrak{a}})=({\mathfrak{a}}^{[p^{e}]})^{[1/p^{e}]}$. For any ideal ${\mathfrak{b}}$ in ${\mathcal{O}}_{X}$ such that ${\mathfrak{a}}^{[p^{e}]}\subseteq{\mathfrak{b}}^{[p^{e}]}$ we get ${\mathfrak{a}}\subseteq{\mathfrak{b}}$ from (4.1). Hence (4.3) implies

(4.5)

$${\mathfrak{a}}\subseteq\tau({\mathfrak{a}}).$$