Depends on in which set are you looking at.
If you are looking in the set of Q (rationals), B doesn't have a supernum in Q.
If you are looking in the set of R (reals), B (and for that matter any bounded set) has a supernum in R. (This is the completeness property of the Reals.
So both books are right. What you missed is the fact that one was talking of B as a subset of Q and other B as a subset of R