I'm reading the Kirk-Goebel's book, "Topics in metric fixed point theory" and I don't get one implication of an equivalence proof. I'm talking about the Lemma 4.1. It sais as follow:
A bounded convex susbet $K$ of a Banach space has normal structure if and only if it does not contain a diametral sequence.
My problem comes when they prove the if implication. They suppose that $K$ contains a diametral sequence $(x_n)_{n\geq 1}$ and then conclude that $S= \operatorname{conv} \{ x_1,x_2,\dots\}$ is diametral, so $K$ has not normal structure.
I've tried to fill the blanks but I'm not capable. Could you guys help me?
Thank you very much.