# Nocedal & Wright — proof of theorem 3.7

On page 47 of Nocedal & Wright's Numerical Optimization, the authors provide a proof for Theorem 3.7 and there is one idea that I didn't understand. The authors prove the following inequality:

$$\|x_k+p_k-x^*\| \le \|x_k+p_k^N-x^*\| + \|p_k-p_k^N\| = O(\|x_k - x^*\|^2) + o(\|p_k\|)$$

and say that a simple manipulation of the above inequality reveals that

$$\|p_k\|=O(\|x_k-x^*\|).$$

Question: What ideas and manipulation are needed to prove that $$\|p_k\|=O(\|x_k-x^*\|)$$ from the above inequality?

Thank you for taking the time to read my question. Any advice is helpful.

• Since the link to the PDF file is broken, I removed it. Commented Jun 15, 2021 at 14:53

We have: $\|p_k\| \le \|x_k -x^*\| + \|x_k + p_k -x^* \|$, and so, $\|p_k\| \le \|x_k -x^*\| + O(\|x_k-x^*\|^2) + o(\|p_k\|)$.
Choose $K$ large enough so that $o(\|p_k\|) \le {1 \over 2} \|p_k\|$, then for $k \ge K$ you have ${1 \over 2} \|p_k\| \le \|x_k -x^*\| + O(\|x_k-x^*\|^2)$, from which the result follows.
• Glad to help! ${}{}$ Commented Oct 7, 2016 at 16:06