I apologize for the indescriptive title, it's the best I could come up with. - Currently I'm working through a paper by Qing Han on 'Nodal sets of Harmonic Functions' and in this paper he uses the following result from his work in 'Singular sets of Solutions to Elliptic Equations'. This result is proven in both works of him. (p. 676 or p. 993)
Suppose $A \subset \mathbb{R}^n$ has the following property: For any $x \in A$ there exists a $j$-dimensional linear subspace $l_x$ such that for any sequence $x_n$ in $A$ with $x_n \longrightarrow x$ we have $$ \text{Angle} <\overline{x_n x}, l_x>\ \longrightarrow 0. $$ Then $A$ is on a countable union of $j$-dimensional Lipschitz graphs.
I struggle with the following proof from the first paper.
For any $x \in \mathbb{R}^n$ let $l_x$ be the corresponding subspace with the property. For $\epsilon > 0$ we define $$C_\epsilon(l_x) = \{y \in \mathbb{R}^n: \text{dist}(y,l_x) < |y|\epsilon\}$$ Due to the limit of the angle above, there exists a constant $r = r(\epsilon,x)> 0$ such that $$ A \cap B_r(y) \subset B_r(y) \cap C_\epsilon(l_x) \text{ for all } y \in A \cap B_r(x). $$ For $\epsilon$ small enough this clearly implies that $A \cap B_r(x)$ is contained in a $j$-dimensional Lipschitz graph. Furthermore, due to the angle property, the graph is even $C^1$.
The proof is short, but there are quite a few things I don't understand. How do I follow the inclusion from the angle property (How does the angle relate to the distance in this case?) More importantly: How does that imply that the left handside is contained in some $j$-dimensional Lipschitz graph? Since $C_\epsilon$ is a set in $\mathbb{R}$^n shouldn't a Lipschitz graph covering this set be of dimension $n$? Or am I missunderstanding the definition of graph dimension?
I am quite desperate at this point and feel like I'm missing some essential especially information especially for the last conclusion regarding the Lipschitz graph. I would appreciate any help.