Assume $M$ is a smooth manifold and $S$ is a subset of $M$ such that each point $p\in S$ has a neighborhood $U \subseteq M$ such that $S\cap U$ is an embedded $k$-submanifold of $U$.
I want to show that from this it follows that $S$ is also an embedded $k$-submanifold of $M$.
Until now I tried to exploit the $k$-slice condition for embedded submanifolds $S\cap U$ of $U$ in order to construct a smooth chart $(V,\phi)$ of $M$ such that $V\cap S$ is a $k$-slice of $V$, i.e. to show that also $S$ satisfies the $k$-slice condition and is therefore an embedded submanifold. But haven't succeeded yet. Can someone help and give me a hint how to approach this problem?
Definition of a $k$-submanifold: A subset $S$ of $M$ is called embedded submanifold of dimension $k$ if for each $p\in S$ there a smooth chart $(U,\phi)$ of $M$ such that $p\in U$ and $U\cap S$ is a $k$-slice of $U$.
Edit: Here are some details on my approach so far. We know that $$\forall p \in S ~\exists U \subseteq \mathcal{U}_M(p):~ U\cap S \text{ embedded }~k\text{-submanifold of } U $$ $$\Rightarrow \forall p \in S ~\exists U \subseteq \mathcal{U}_M(p)~\forall q\in U\cap S~\exists (V,\phi) \text{ smooth chart of } U \text{ with } q\in V:$$ $$ U\cap S\cap V ~k\text{-slice of } V $$ But since $V$ is a coordinate neighborhood of $U$, i.e. $V\subseteq U$, and as noted in the comments $V$ is also a coordinate neighborhood of $M$, we have $$\forall p \in S ~\exists U \subseteq \mathcal{U}_M(p)~\forall q\in U\cap S~\exists (V,\phi) \text{ smooth chart of } M\text{ with } q\in V:$$$$ S\cap V ~k\text{-slice of } V $$
But since $p\in U\cap S$ we also find a smooth chart $(V,\phi)$ of $M$ with $p\in V$, such that $S\cap V$ is a $k$-slice of $V$. Which means that by the definition above we know that $S$ is an embedded $k$-submanifold of $M$.