Local immersion theorem

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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)