Proving compact $m$-manifolds can be embedded in $\mathbb{R}^{nm}$ for some cover $\{(B_i,\varphi_i)\}_{i=1}^n$.

Question

When a smooth $m$-manifold $M$ is compact, choose $\{(B_i,\varphi_i)\}_{i=1}^n$ a covering by coordinate balls. Use bump functions $\chi_i$ so that ${\chi_i}|_{B_i}=1$. Define \begin{equation} F(p)=(\chi_1 \varphi_1(p),\dots, \chi_n \varphi_n(p))\in\mathbb{R}^{nm}. \end{equation} We can show this is a smooth injective immersion $M\to\mathbb{R}^{nm}$, and hence an embedding since $X$ is compact. My question is, how does one show this is an immersion?

Answer 1

All you have to do is show that the derivative is injective at any $p$. So WLOG suppose $p \in B_1$, then in that neighbourhood, $F(q) = (\varphi_1(q),...,\chi_n \varphi_n(q))$ since $\chi_1 = 1$ there. We see that $dF_p$ has an invertible $m \times m$ sub-matrix corresponding to $\varphi_1$ since it is a diffeomorphism, and since the matrix is $n \times m$, this makes it an immersion (note that you need $n > m$, so you need a sufficiently "fine" partition of unity).