= Existence of a Wolfe step
{c}
Suppose $f$ is <continuously differentiable>, $s$ is a <descent direction>, and $f(x+\alpha s)$ is bounded below for $\alpha\geq0$. A positive <step size> satisfying the weak <Wolfe conditions> exists. To prove this, put $\phi(\alpha)=f(x+\alpha s)$, $d=\phi'(0)<0$, and $h(\alpha)=\phi(\alpha)-\phi(0)-\beta_1\alpha d$. The <Armijo condition> proof shows $h<0$ just to the right of zero. Boundedness below gives $h(\alpha)\to+\infty$. Consequently there is a first positive zero $A$, with $h\leq0$ on $[0,A]$. By the <mean value theorem>, some $t\in(0,A)$ satisfies
$$
\phi'(t)=\frac{\phi(A)-\phi(0)}A=\beta_1d>\beta_2d.
$$
At $t$, $h(t)\leq0$ gives the <Armijo condition>, and the displayed derivative gives the weak <Wolfe conditions>.
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