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.
Best Answer
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.