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.