First I think that you have notational problems.

You have .

I think that you want: .

From the diagram 11 precedes every element except 12. In order to havea smallest element it must precede every element. Likewise, there is no element that is preceded by every element, so there is no greatest element.

Now consider the set A. Both 1 & 11 precede every element of A. Therefore, each of 1 & 11 is a lower bound of A. But 1 is the inf(A) because it is the greatest lower bound.

Now consider the set C. Both 8 & 10 are preceded by each element of C (and yes 8 is said to precedes itself). But 8 is the sup(C) because it is the least upper bound.