I read Theorem 4.25 in Lee's Smooth Manifolds book. This theorem states that a smooth map $F:M\rightarrow N$ is a smooth immersion if and only if it is locally a smooth embedding (let's ignore in this post his discussion of manifolds with boundary).
The harder direction is smooth immersion $\Rightarrow$ locally smooth embedding. We need to prove that locally $F$ is a homeomorphism into its image. When I tried to prove it by myself, I used the constant rank theorem and said: locally (in coordinates) $F$ looks like $(x_1,\dots,x_m)\mapsto (x_1,\dots,x_m,0,\dots,0)$, which is a homeomorphism into its image, and that's it (I wasn't certainly sure it works).
Then I read Lee's proof, and he used the constant rank theorem only to say that locally, on an open $U\subseteq M$, the function $F$ is injective. He then took an open precompact $V\subseteq U$, and stated that $F|_{\overline{V}}$ is a homeomorphism into its image by the closed map lemma. It follows that $F|_V$ is a homeomorphism into its image since it is a restriction of such a function.
I was surprised to read his proof, since I thought that knowing that locally the coordinate representation of $F$ is a map which is a homeomorphism into its image is enough. I thought that Lee's proof is somewhat unnecessarily long and complicated.
Is my proof wrong maybe? Or maybe Lee thought that his proof is shorter? Or maybe it sheds light on other techniques to educate the readers?
If somebody wants, my formal proof, which is supposed to be shorter: Let $p\in M$. There are charts $p\in (U,\varphi)$, $F(p)\in (V,\psi)$ such that $\hat f := \psi \circ F \circ \varphi^{-1} : \varphi(U)\to \psi(V)$ is a homeomorphism into its image. We define $\tilde f$ to be $\hat f$ with the range restricted to $\operatorname{Im}(\hat f)$, so $\tilde f : \varphi(U)\to \operatorname{Im}(\hat f)$ is a homeomorphism. So the function $F|_U$, with its range restricted to its image, is the composition $\psi^{-1}|_{\operatorname{Im}(\hat f)} \circ \tilde f\circ \varphi$, which is a homeomorphism as a composition of homeomorphisms.