# Math Help - Supremum of Confusion

1. ## Supremum of Confusion

I got from two different books a set $B= \{r \in \mathbb{Q}: 0 \leq r \leq \sqrt{2}\}$.

One book says $B$ does not have $sup (B)$ and $\sqrt{2}\not = sup(B)$ because $\sqrt{2} \not \in B$.

The other book says $sup(B) = \sqrt{2}$. Although $\sqrt{2}\not \in B$, there are rationals in the set arbitrarily close to $\sqrt{2}$.

Who is right?
The first book is a textbook, and the second is a supplement.

2. 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