$\newcommand{\pl}{\partial}$ $\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be smooth compact, connected, oriented manifolds of the same dimension. $\M$ can have a boundary.
Let $\,f_n:\M \to \N$ be a sequence of smooth orientation-preserving embeddings, which converges uniformly to a smooth immersion $f:\M \to \N$.
Is it true that $f|_{\M^o}$ is injective? ($\M^o$ is the interior of $\M$).