pf. Suppose that x and y are both the greatest lower bound. Since x and y are greatest lower bounds, then x and y are lower bounds. Since x is a lower bound and y is a greatest lower bound. We must have (y,x)$\in$ R. Likewise, since y is an lower bound and x is a greatest lower bound, we must have xRy. From xRy and yRx, we conclude that x=y by antisymmetry. Thus if it exists, inf(B) is unique
I followed the steps out of my textbook, but I am not sure if this is correct. Can someone give me some guidance if I proved this theorem correctly. Thanks. inf = infimum(greatest lower bound)