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 $\iota(U_0) \subseteq V_0$.
I am not sure if this is related to isometric coordinate representation, in fact I don't think we need the notion of metric here. For $U$ open in $S$ and $V$ open in $M$, we have coordinate representation of the inclusion map being $(x^1,\ldots,x^k)\rightarrow (x^1,\ldots,x^k,0,\ldots)$. Since $U$ and $V$ are open, we can choose a $k$ dimensional ball of radius $\epsilon_1$ contained in $U$, and a $n$ dimensional ball of radius $\epsilon_2$ in $V$, centered at the point. We simply choose $\epsilon={\rm min}\{\epsilon_1,\epsilon_2\}$. Let $U_0$ be the $k$ dimensional ball of radius $\epsilon$ which is contained in $U$, and $V_0$ be the $n$ dimensional ball of radius $\epsilon$ which is contained in $V$. By the inclusion map, $U_0$ is mapped to identically the same ball which is contained in $V_0$, because they have the same radius just different dimension, like a disk is contained in a 3D ball. We don't need metric, we just need to know each point in the ball is identified with a point in the manifold.