Supremum of a set, relations and order

[WWGD]

TL;DR Summary: prove that a supremum for a set doesn't exist; relations, total order and partial order

Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number thats positive
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I also did some reading on Cartesian product, relations, total order and partial order.
So a Cartesian product AxB is just ordered pair (a,b) where a is an element of A and b is an element of B right.
And a relation is just a subset of the Cartesian product.
Now total and partial orders.
Total order is denoted by < and partial order is denoted by <=? I’m a bit unsure about these, please correct me if I’m wrong.
And a total order relation requires four things:
Reflexive, anti-symmetric, transitive and comparability? I’m a bit unsure what comparability is though.
And for partial order relation I think it just needs to be reflexive, anti-symmetric and transitive?

The product $A \times B$ is not a pair $(a,b)$ but rather the collection of _all_ pairs $(a,b)$ for $a \in A, b\in B$.