I am just puzzled by an observation I made on undergraduate texts in Real Analysis. The concepts of supremum and infimum are not particularly easy to grasp for students who are fresh to the subject, and it is surely a lot easier for them to understand the maximum and minimum of a set (including the fact that they may not exist) and of upper bounds. All one has to do is point out intervals such as $[0,2]$ and $[0,2)$ and talk about the number 2 and the number 3. So why not define the least upper bound of a set $S$ (in an ordered field) as the minimum (it it exists) of the set of all upper bounds for $S$?
I have examined a large number of undergraduate texts for Real Analysis, and I have never found this simple definition, or the word "minimum" even mentioned. Most of the them will define it using the much more difficult way "it's an upper bound, and for any number strictly less than it, there will be some etc. etc. ". A few will say: it is an upper bound $M$ such that if $x$ is an upper bound, then $M\leq x$. Again, why not say, "it's the minimum of the set of all upper bounds"? Some will just say "it's the least upper bound" but even this is a change of language, from minimum to least, and in any case they use "least upper bound" as just the name for the concept, then going on to define it using the convoluted way.
I do not mean to post this as a topic of discussion, only as a genuine question on whether I am missing something important that will explain why it is a good idea to define supremum that way.