Let $M$ be a compact $n$-dimensional differentiable manifold, and $f:M\rightarrow \mathbb{R}^n$ be a differentiable mapping.
Prove that there exists some point $p\in M$ such that $f$ is singular at $p$.
Here '$f$ is nonsingular at $p$' means that $df_p:T_pM\rightarrow T_{f(p)}N$ has rank $n$, and '$f$ is singular at $p$' means that $df_p:T_pM\rightarrow T_{f(p)}N$ has rank below $n$.
Here is something I have tried: Suppose that $f$ is nonsingular at each $p\in M$. Use Inverse Function Theorem, we could show that for each $p\in M$ there is a neighborhood $U_p$ of $p$ such that $U_p$ and $f(U_p)$ are homeomorphism. So $(U_p,f)_{p\in M}$ is a coordinate neighborhood of $M$. But I don't know how to lead to a contradiction.