I am trying to understand a proof in Lee - Introduction to Smooth Manifolds. I have exactly the same question as was asked here: A submanifold is embedded iff it satisfies the local k-slice condition. The answer there says the containment $\iota(U_0)\subseteq V_0$ is clear, but I cannot see it. It is clearly true in coordinates, since for $u \in U_0$ we have that $\iota(u) = (u^1, \dots, u^k,0, \dots, 0) \in \iota(U_0)$ is also an element of $V_0$. However, I cannot see how to conclude the result.
I am in particular worried about the fact that all balls in Euclidean space are diffeomorphic to each other. Therefore the fact that $U_0$ and $V_0$ both have radius $\varepsilon$ in local coordinates does not seem sufficient to guarantee that $\iota(U_0) \subseteq V_0$.