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 $\lambda \to 0$.