Corollary 6.17 in Lee's ISM states that certain smooth maps from manifolds to $\mathbb{R}^n$ can be "uniformly approximated by embeddings." Problem is, I don't know what uniform approximation means. My initial guess at a definition was the following:
Let $f : X \to Y$ be a map where $X$ is a set and $Y$ is a metric space, and let $\mathcal{C} = \{\phi_\alpha\}$ be a collection of functions $\phi_\alpha : X \to Y$. Then $f$ can be uniformly approximated by $\mathcal{C}$ if for all $\delta > 0$, there exists an $\alpha$ such that $|f(x) - \phi_\alpha(x)| < \delta$ for all $x \in X$.
I don't think this definition is right anymore, since Lee claims that projections in the directions of two distinct vectors can be arbitrarily close. This doesn't work with my tentative definition because if $v,w$ are two vectors in $\mathbb{R}^n$ that are not in $\mathbb{R}^{n-1}$ with the standard embedding of $\mathbb{R}^{n-1}$ into $\mathbb{R}^n$), then for all $\delta > 0$, we can find a point $x$ such that $|\pi_v(x) - \pi_w(x)| > \delta$ (where $\pi_v,\pi_w : \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the projections in the directions of $v$ and $w$).
I see two possibilities here. Either (i) my definition actually is correct and the contradiction explained above can be recitifed by noting that the domains considered in the theorem are compact and thus bounded, or (ii) I guessed the wrong definition. How do we rigorously treat the idea of uniform approximation?
Your possibility (i) is correct. Notice that Corollary 6.17 applies only to compact manifolds. If $M$ is a compact manifold embedded in $\mathbb R^n$, then $\sup_{x\in M} |\pi_v(x) - \pi_w(x)|$ can be made as small as desired by choosing $v$ sufficiently close to $w$.