# Proof of Criteria for Local Identification in Rothenberg (1971)

My question is regarding Theorem 1 (page 579) of Rothenberg (1971). It is associated with four assumptions given on the same page. But, I only have a question about a single step of the proof, so I don't think the entire context is necessary.

Definition 3: A parameter point $$\alpha^0$$ is said to be locally identifiable if there exists an open neighborhood of $$\alpha^0$$ containing no other $$\alpha$$ in $$A$$ which is observationally equivalent
...
Assumption I: The structural parameter space $$A$$ is an open set in $$\mathbb{R}^m$$
...

Under Assumptions I-V given in the paper, we have

Theorem 1: Let $$\alpha^0$$ be a regular point of the matrix $$R(\alpha)$$. Then $$\alpha^0$$ is locally identifiable if and only if $$R(\alpha^0)$$ is nonsingular.

In the proof for one direction of the iff statement, the author begins by supposing $$\alpha^0$$ is not locally identifiable. This means there exists an infinite sequence of vectors $$\{\alpha^1, \alpha^2, ..., \alpha^k, ...\}$$ approaching $$\alpha^0$$ such that $$g(y, \alpha^k) = g(y, \alpha^0)$$ for all $$y$$ and each $$k$$.'' Then, the author later claims that

$$d_i^k = \frac{\alpha_i^k - \alpha_i^0}{|\alpha^k - \alpha^0|}$$

has a limit point $$d$$ on the unit sphere, and, as $$\alpha^k \to \alpha^0$$, $$d^k$$ approaches $$d$$.''

This does not make sense; I agree that there is a limit point but not that there is a limit.. As a counterexample, suppose $$A = \mathbb{R}$$, $$\alpha^0 = 0$$, and the infinite sequence was

$$\{\alpha^k \}_{k=1}^\infty = (-1)^n \cdot (1/n)$$

This sequence is valid because it approaches $$\alpha^0 = 0$$, while we can simply assume that the $$g(y, \alpha^k) = g(y, \alpha^0)$$ condition holds for all $$y,k$$ for the sake of contradiction. Using this sequence, we get

$$d^k = \frac{\alpha_i^k - \alpha_i^0}{|\alpha^k - \alpha^0|} = \frac{(-1)^n \cdot (1/n)}{(1/n)} = (-1)^n$$

which has limit points but no limit.

Is there an explanation for why the author assumes such a limit exists? I did not catch any other conditions on $$\alpha$$ in the paper that could make this step in the proof work.