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 $df_p$ 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:

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