Hi,
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.
This shows that is a lower bound. Now, you must show that it is greatest among all lower bounds. So, suppose there is a lower bound of A that is greater and use this to get an upper bound of -A which is smaller than sup(-A). This contradicts the leastness of sup(-A).
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.