Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 7 / v / Solution

The Armijo condition with $p_k=-g(x_k)$ gives $f(x_k)-f(x_{k+1})\ge \beta {\alpha }_k\Vert g(x_k){\Vert }_2^2$. Summing these inequalities telescopes toIf $f\ge f_{\inf }$, these partial sums of nonnegative terms are bounded above by $(f(x_0)-f_{\inf })/\beta$. The resulting series converges, so its terms tend to zero: $\boxed{{\alpha }_k\Vert g(x_k){\Vert }_2^2\to 0}$.