Finite injective immersion from a separating family

Let $M$ be a Hausdorff, second-countable smooth manifold of dimension $n$, and let $\{f_k\}_{k=0}^\infty$ be a countable sequence of smooth functions with compact support that separate points and separate tangent vectors. Then there exists a finite integer $N$ and a choice of $N$ functions from the sequence (possibly repeated or linearly combined) that form a smooth injective immersion $e : M \to \mathbb{R}^N$.

The proof relies on a Baire category argument or a transversality argument. One considers the infinite-dimensional map $F : M \to \mathbb{R}^\infty$ given by $F(x) = (f_0(x), f_1(x), \dots)$. Since the functions separate points and tangent vectors, $F$ is an injective immersion. Because the functions have compact support and $M$ is second-countable, one can show that for a generic finite-dimensional projection $\pi : \mathbb{R}^\infty \to \mathbb{R}^N$, the composition $\pi \circ F$ remains an injective immersion, provided $N$ is sufficiently large (e.g., $N \ge 2n+1$).