Let $M$ be a smooth manifold. I am trying to prove that $M$ admits a vector field with only finitely many zeros.
This will follow if we can find a function $f : M\rightarrow \mathbb R$ such that $df$ has only finitely many zeros, but I cannot find such a function with this property either. My initial idea was to try to embed $M$ in $\mathbb R^N$ for some $N$ and look at $x\mapsto u \cdot x$ for fixed $u\in \mathbb R^N$, but I could not find a way to prove that there must be a $u$ such that the differential of this map has only finitely many zeros.
Does anyone have an elementary construction of such a vector field (or function)?