How $||\nabla f(x) - \nabla f(\overline{x})||\le L|x-\overline{x}|$ implies $(\nabla f(x) -\nabla f(\overline{x}))^Tp_k\le L|x-\overline{x}||p_k|$

Question

I'm trying to understand how the lipschitz condition (3.13) implies the last equation. The rest is specific for optimization and is not needed to understand the question but I leave it here for context. The book is from Nocedal.

As far as I understood, the Lipstichtz condition is on the norm. We can only say what's in 3.13. However, the least equation of the image is not on the norm. How do I prove such implication? (the right side is just $L||x_k+\alpha_k p_k - x_k||$)

Not necessary for the question but just to inform:

If you want to know more about it, the conditions 3.6b and 3.1 are, respectively:

$$\nabla f_{k+1}^Tp_k\ge c_2\nabla f_k^Tp_k$$ $$x_{k+1} = x_k + \alpha_k p_k$$

Answer 1

\begin{align} (\nabla f_{k+1}-\nabla f_k)^T p_k &\le \|\nabla f_{k+1}-\nabla f_k\|\|p_k\|, \text{by Cauchy-Schwarz}\\ &\le L \|x_{k+1}-x_k\|\|p_k\| \text{, by Lipschitz} \\ &=L\|\alpha_k p_k\|\|p_k\|\\ &= L\alpha_k \|p_k\|^2 \end{align}