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?