Upper Bound vs. Least Upper Bound

Question

I am reading Rudin's Principles of Mathematical Analysis in order to prepare for my first course in Real Analysis I intend to take this fall. The book just defined what an upper bound is and then defined supremum/ least upper bound as:

Suppose $S$ is an ordered set, with $E$ as a subset of $S$, where $E$ is bounded above. Suppose there exists an $\alpha \in S$ with the following properties:

A) $\alpha$ is an upper bound of $E$.

B) If $\gamma < \alpha$ then $\gamma$ is not an upper bound of $E$.

I do not understand the difference between upper bound and least upper bound. If someone could explain the difference between the two and possibly provide an example, it would be much appreciated. Thanks.

Answer 1

Every least upper bound is an upper bound, however the least upper bound is the smallest number that is still an upper bound. Example: Take the set $(0,1)$. It has $2$ as an upper bound but clearly the smallest upper bound that the set can have is the number $1$ and hence it's the least upper bound.

Answer 2

Maybe you like this definition better?

Let $A$ be nonempty. We say that $\alpha$ is the least upper bound of $A$ if

$(1)$ It is an upper bound of $A$, that is, if $x\in A$; then $x\leq \alpha$.

$(2)$ If $\beta$ is any other upper bound $\alpha\leq \beta$. That is, $\alpha$ is the least of all upper bounds of $A$.

As you can see the l.u.b. has the unique property $(2)$. Why unique? Because if $\gamma$ is another l.u.b., by definition, we must have both $\alpha\leq \gamma$ and $\gamma\leq \alpha$, but this means we must have $\alpha=\gamma$. So l.u.b.s when they exist, are unique.