Suppose that $(k,\lvert\mathord{\cdot}\rvert)$ is discretely valued. Let $\phi$ be an asymptotic Fubini-Study metric on $L$. Then for any $\epsilon>0$, there exists $n_{Y}\in\mathbb{N}$ such that for any $n\geq n_{Y}$ and any $t_{n}\in V_{n}(L|_{Y})$, there exits $s_{n}\in V_{n}(L)$ such that $s_{n}|_{Y}=t_{n}$ and