I'm reading the proof of the standard result on KKT conditions in Nocedal and Wright's textbook (2nd edition). In definition 12.2, they define the tangent cone of the feasible region $\Omega$ at $x$, denoted by $T_\Omega(x)$. Generically, this cone should be same as the cone $\mathcal{F}(x)$ of linearized feasible directions, defined in Definition 12.3.
An important point is that $T_\Omega(x)$ does not depend on the algebraic specification of $\Omega$, while $\mathcal{F}(x)$ does. Simple examples show that they are not always the same when the gradients of the constraint functions at $x$ are not linearly independent.
In their Lemma 12.2, they intend to show that when the gradients of the constraint functions are linearly independent, then the two cones are the same. It seems like a straightforward application of the implicit function theorem, but in the last line of their proof, on Page 325, they write: $\frac{z_k-x^\ast}{t_k} = d + o(\frac{\| z_k - x^\ast\|}{t_k})$. And from this they conclude that $\frac{z_k-x^\ast}{t_k} \rightarrow d$.
Nowhere do they explain why the quotient $\frac{\| z_k - x^\ast\|}{t_k}$ should go to zero. Am I missing a simple point in the proof?
