= Solution
Set $p=x^{k+1}-x^k=\alpha^ks^k$ and $y=\nabla f(x^{k+1})-\nabla f(x^k)$. The <Quasi-Newton secant equation> is $B^{k+1}p=y$. If $B^{k+1}$ is a <positive-definite symmetric matrix> and $p\ne0$, it follows that
$$
p^Ty=p^TB^{k+1}p>0.
$$
The weak curvature part of the <Wolfe conditions> ensures precisely this necessary condition:
$$
p^Ty=\alpha^k\bigl(\nabla f(x^{k+1})^Ts^k-\nabla f(x^k)^Ts^k\bigr)\geq\alpha^k(\beta_2-1)\nabla f(x^k)^Ts^k>0.
$$
The last product is positive because $\alpha^k>0$, $\beta_2-1<0$, and $s^k$ is a <descent direction>. This explains why a <Quasi-Newton method> often uses the <Wolfe conditions> rather than only the <Armijo condition>.
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