Solution
= Solution
The <Armijo condition> with $p_k=-g(x_k)$ gives $f(x_k)-f(x_{k+1})\geq\beta\alpha_k\|g(x_k)\|_2^2$. Summing these inequalities telescopes to
$$
\boxed{f(x_0)-f(x_{j+1})\geq\beta\sum_{k=0}^j\alpha_k\|g(x_k)\|_2^2.}
$$
If $f\geq f_{\inf}$, 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: $\boxed{\alpha_k\|g(x_k)\|_2^2\to0}$.