A question about a proof in nonlinear programming book

Question

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof the limit point is assumed to have a nonzero gradient, but it seems this assumption is not used in the rest of the proof. It seems even if the limit point's gradient is zero, the contradiction at the end of the proof still holds.

I post a copy of the proof below: Can any one help? Thanks.

Answer 1

I see I missed gradient related is defined for nonstationary points.