This is exercise 10 on page 19 in Guillemin/Pollack Differential Topology.
Let $f : X \to Y$ be a smooth map which is one-to-one on some compact submanifold $Z$ of $X$. Moreover, let $df_p$ be nonsingular for all $p \in Z$. Then there exists some neighbourhood $U$ of $Z$ in $X$ such that $f\vert_U$ is one-to-one.
Towards a contradiction, assume that $f$ is not injective on any neighbourhood of $Z$. Now the hint states that we should construct two sequences $(x_n)$ and $(y_n)$ in $X$ converging to some $z \in Z$ such that $f(x_n) = f(y_n)$. So since Guillemin/Pollack only work with $\mathbb{R}^n$, I thought we could consider the neighbourhoods $$U_\varepsilon := \{x \in \mathbb{R}^n : \text{dist}(x,X) < \varepsilon\}$$ Now choosing $\varepsilon = 1/n$ for $n \in \mathbb{N}$ yields two sequences $x_n$ and $y_n$ such that $f(x_n) = f(y_n)$. But I do not quite see how these sequences should converge (or at least subsequences) to a common point $z \in Z$. So my questions are:
- Is my choice of the sequences right?
- Is it possible to generalize this result to arbitrary manifolds, i.e. not involving Euclidean spaces? I mean the definition of $U_\varepsilon$ can be adapted by equipping $M$ with a Riemannian metric.