Real Analysis – The Neat Banach Space Concept

In this article, the authors used the notion of ordered Banach space.

Definition: Let $ mathcal E $ to be a Banach space with the standard $ left | . right | $, whose positive cone is defined by $ K = {x in mathcal E : : x geq 0 } $. then $ ( mathcal E, left |. right |) $ is now a partially ordered Banach space with the order relationship $ sqsubseteq $ induced by the cone K $.

and they used this result:

Theorem:Let $ ( mathcal E, left |. right |, sqsubseteq) $ to be an orderly space of Banach, whose positive cone K $ Is normal(?). Let $ (u_n) _ {n in mathbb N} $ a monotonous sequence (let's say $ u_n sqsubseteq u_ {n + 1} $), such as $ {u_n} $ has a convergent subsequence. Then the whole sequence $ u_n $ is convergent?

I have two questions here:

$ 1. $ What is it? $ a sqsubseteq u _ {n + 1} $ means, is there an interpretation of that?

$ 2. $ I wonder why the whole sequence $ u_n $ is convergent?

Please share your thoughts and thank you in advance!