Definition: An embedding of manifolds is a differentiable map $f:N^n\hookrightarrow M^m$ which has an injective pushforward. An immersion of manifolds $f:N^n\looparrowright M^m$ is a differentiable map which is locally an embedding. That is, for every $x\in N$, there is an open $V\subset N$ such that $x\in V$ and $f|_V: V \hookrightarrow M$ is an embedding.
I am attempting to prove the following lemma; my attempt is below.
Lemma: Any injective immersion of closed manifolds $f:N\looparrowright M$ is an embedding.
Proof: Take a cover of $N$ by open sets $\{U_i\subset N\}_{i\in I}$ such that $f|_{U_i}:U_i\hookrightarrow M$ is an embedding. Take a finite subcover $\{U_i'\}_{i=1}^k$ and let $\{\varphi_i:N\to [0,1]\}_{i=1}^k$ be a partition of unity subordinate to $\{U_i'\}$. Then for any $x\in X$ \begin{equation} D_xf= \sum_{i=1}^k \varphi_i(\delta_i(x)) \delta_i(x_i) \end{equation} Where $\delta_i(x)=D_x f|_{U_i'}$ if $x\in U_i'$ and $0$ otherwise. This is injective because each $D_xf|_{U_i'}$ is injective.
I know this is incorrect but I'm wondering if it could be fixed (I'd prefer an answer using partitions of unity - if that is the right thing to use here).