Let $A$ is a submanifold of $\mathbb{R}^n$ with $\dim A<n$, $M^m$ is a differential manifold. Suppose $n\geq 2m+1$. Prove that there exists an embedding $h: M\rightarrow \mathbb{R}^n$ such that $h(M)\cap A=\phi$.
I just learned the Whitney embedding theorem. Intuitively, $\dim A<n$ implies $A$ is of measure $0$ in $\mathbb{R}^n$, so the image of $h$ is not likely to "touch" $A$.
Can the proposition be proved by modifying the proof of Whitney embedding theorem? Appreciate any help
Pick an open ball $B(x, r) \subset \mathbb R^n\setminus A$. Since $B(x, r)$ is diffeomorphic to $\mathbb R^n$, it admits an embedding of $M$ by Whitney.