the whole problem and solution is at: Solutions – Baby Rudin – 1.5 (work in progress) « DaFeda's Blog. I'm sorry for posting an external link, just trying to avoid writing it all again. However, I do not think you even need to go there to answer my question.
In my proof I write;
If then . Since for all this would imply that
, which is a contradiction since for all .
Can I use and in such a way? I have three different solutions available to me from different people, so I am not looking for another way of solving this.
Hope someone has the time to take a look! Thanks.
In general, to show something is a sup/inf, you need to check two things: (1) it is a upper/lower bound and (2) it is (nonstrictly) smaller/larger than any particular upper/lower bound.