Local immersion theorem

Statement

Let $M$ and $N$ be differential manifolds of dimensions $m<n$. Let $f:M\to N$ be a differentiable map between them. Let $p\in M$ be a point such that $f$ is an immersion at $p$ -- in other words, the induced map $d{f}_p$ is injective. Then, there exists a neighbourhood $U\ni p$ in $M$ and $V\ni f(p)$ in $N$, and a choice of coordinate charts, such that the restriction of $f$ to $U$, viewed using those coordinate charts, is a map of the form:

$(x_1,x_2,\dots ,x_m)\mapsto (x_1,x_2,\dots ,x_m,0,0,\dots ,0)$