I tried looking through other questions but couldn't find this one answered. I'm working through Lax and Terrell's 'Calculus with Applications'.
The book asks to state whether the following intervals have GLBs and LUBs and if so, what they are: $(8,10)$, $(8,10]$.
My initial thought was that an open interval can't have a LUB or GLB as they don't include the end points and thus there is always a number arbitrarily closer to the end point that doesn't quite 'reach' it. However, I've also seen some discussion such that, e.g., the LUB of both half-open and open intervals are the same given that $9.99999… = 10$.
Help appreciated here as I want to understand this before proceeding!
Best Answer
Every bounded non-empty subset of $\Bbb R$ has a greatest lower bound and a lowest upper bound. In the case of $(8,10)$ and of $(8,10]$, these are $8$ and $10$ respectively. The fact that $10$ belongs to the second interval but not to the first one changes nothing. And the fact that $9,99999999\ldots=10$ also changes nothing.