Finite-dimensional injective immersion for a noncompact manifold

Let $M$ be a noncompact, Hausdorff, second-countable smooth manifold of dimension $n$. Then there exists a finite integer $N$ and a smooth injective immersion $e : M o \mathbb{R}^N$. An immersion is a smooth map whose differential is injective at every point. The existence of a finite-dimensional injective immersion is the core step in the weak Whitney embedding theorem for noncompact manifolds, reducing the infinite-dimensional space of smooth functions to a finite-dimensional Euclidean space via generic linear combinations of a countable partition of unity.