# Convergence proof on Newton's method from Boyd & Vandenberghe's book on convex optimization

From Boyd & Vandenberghe's book, page 491:

Applying the Lipschitz condition,we have \begin{aligned} \left\|\nabla f\left(x^{+}\right)\right\|_{2} &=\left\|\nabla f\left(x+\Delta x_{\mathrm{nt}}\right)-\nabla f(x)-\nabla^{2} f(x) \Delta x_{\mathrm{nt}}\right\|_{2} \\ &=\left\|\int_{0}^{1}\left(\nabla^{2} f\left(x+t \Delta x_{\mathrm{nt}}\right)-\nabla^{2} f(x)\right) \Delta x_{\mathrm{nt}} d t\right\|_{2} \\ & \leq \frac{L}{2}\left\|\Delta x_{\mathrm{nt}}\right\|_{2}^{2} \\ &=\frac{L}{2}\left\|\nabla^{2} f(x)^{-1} \nabla f(x)\right\|_{2}^{2} \\ & \leq \frac{L}{2 m^{2}}\|\nabla f(x)\|_{2}^{2} \end{aligned} to get the inequality assumption (9.33), which is:

## $$\frac{L}{2 m^{2}}\left\|\nabla f\left(x^{(k+1)}\right)\right\|_{2} \leq\left(\frac{L}{2 m^{2}}\left\|\nabla f\left(x^{(k)}\right)\right\|_{2}\right)^{2}$$

I dont understand how he get the first two lines of the derivation?

My understanding is that $$\nabla f\left(x^{+}\right)$$ equals $$\nabla f\left(x+\Delta x_{\mathrm{nt}}\right)$$ already when t=1, so where do the rest of two terms come from? please anyone help me, really appreciate.