How can a finite set not have a maximum or minimum? Consider a set of real numbers such that $0\le x&lt;1$. My book (Tom Apostol) says that this set do not have any maximum. How is it possible? Isn't the number just below $1$ the maximum of the set? Isn't that the least upper bound (supremum)? Why is $1$ the supremum?
The set of real numbers in $[0,1)$ isn't finite, it is bounded.
There is no real number "just below" 1. Give any real number $\alpha &lt; 1$
and $\alpha + \frac{(1-\alpha)}{2}$ is also strictly less than 1, and strictly greater than $\alpha$.
For any real number a which is in S.
There exists another number b which is larger than a: b=(a+1)/2.
So there is no maximum.