Chris Leary

I'm going to make this one comment and then be done. It has been established that $1,2,3,4$ are all upper bounds for your set. According to Rudin's definition, $4$ cannot be the least upper bound because there are upper bounds that are less than $4,$ namely $1,2,3.$ For a similar reason neither $3$ nor $2$ can be upper bounds. Of all possibilites, only $1$ satisfies the properties that it is an upper bound and that no smaller number is an upper bound. Walk away from this for an hour or so, come back with a refreshed mind, and think carefully about what the two definitions are actually saying.

@cag51 - I have had experience with math majors who could not add fractions. I also had a math major who wanted to be a high school math teacher and didn't undertand why they had to study polynomials in abstract algebra, claiming they "already knew how to factor." Nonetheless, the student could not factor simple quadratics when test time came. I second Dilworth's assessment and have become a cynic in this regard.

I had a short conversation with the Graduate Dean before handing in my dissertation. It was quite reassuring. The one remark that has stuck with me was "I'm not going to go through your thesis with a ruler measuring the margins." As many others have said, relax and enjoy. Congratulations.

Keep on keeping on. I would think you should be able to contact your adviser at any time you think you need to ask a question or seek advice. Generally, your supervisors are busy people. As long as you don't constantly contact them, they should understand. As others have said, you seem to be progressing nicely with little supervison. Remember that the Ph.D. process is designed to produce independent scholars.