A question about a proof of projection theorem

Question

I'm reading a proof of projection theorem on $\mathbb R^n$

I'm unable to understand how the author infers $\langle z-x^{*}, x^{*}-x\rangle \geq 0$ from $\lambda^{2}\|x-x^{*}\|^{2}+ 2 \lambda \langle z-x^{*}, x^{*}-x\rangle \ge 0$.

Please elaborate more on this point! Thank you so much!

Answer 1

Divide by $\lambda$ and take limit as $\lambda \to 0$.

Answer 2

Maybe a not so "subtle" argument helps too.

The proof implicitly uses the following fact, which can easily be shown by contradiction:

• If a number $a$ satisfies for all $\epsilon > 0$ the inequality $a \leq \epsilon$, then $a \leq 0$.

Now, for $\lambda >0$ you have $$\lambda^{2}\|x-x^{*}\|^{2}+ 2 \lambda \langle z-x^{*}, x^{*}-x\rangle \ge 0$$ $$ \underbrace{\Leftrightarrow \langle z-x^{*}, x - x^{*}\rangle}_{a} \leq \underbrace{\lambda\|x-x^{*}\|^{2}}_{\epsilon}$$

Hence, $$\langle z-x^{*}, x - x^{*}\rangle \leq 0$$