Existence of subspace disjoint from manifold

Question

Suppose $M$ is a differentiable $m$-submanifold of $\mathbb{R}^n$, where $m < n$. For $s < n − m$, I would like to show there is an $s$-dimensional affine subspace $A$ of $\mathbb{R}^n$ such that $A\cap M=\emptyset$. Also, what happens for $s=n-m$?
I am not really sure where to start. Intuitively, I can understand that for instance, if you have a surface in $\mathbb{R}^3$, say, then you can find a curve which does not intersect that surface.