 Solution

Fix an iteration and abbreviate $x=x^k$, $s=s^k$, $d=\nabla f(x{)}^{T}s$. Since $s$ is a descent direction, $d<0$. The directional derivative givesChoose $\bar{\alpha }>0$ so that $|o(\alpha )|\le \frac{1}{2}(1-{\beta }_1)(-d)\alpha$ for $0<\alpha \le \bar{\alpha }$. ThenThis is the Armijo condition for every such positive step size, and $\alpha =0$ gives equality. In particular, a backtracking line search terminates after finitely many reductions.