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So I was reading through "Elementary Real and Complex Analysis" by Georgi E. Shilov (reading the first chapter on Real Numbers and all that "simple" stuff like the field axioms, a bit of set stuff, etc.).

Anyways, so while I was reading, I ran into something I couldn't understand... the least upper bound axiom..

It says "A set E (is a subset of) R is said to be bounded from above if there exists an element z (is an element of) R such that x (is less than or equal to) z for every x (is an element of) E, a fact expressed concisely by writing E (is less than or equal to) z. Every number z with the above property relative to a set E is called an upper bound of E. An upper bound zo of the set E is called the least upper bound of E if every other bound z of E is greater than or equal to zo."

OK, so let me get this straight... does this imply anything along the lines of zo being the (numerically) greatest element of E as a subset of R such that every other element R has that E doesn't have is greater than E? (I'm guessing R means all reals here)

So, I continued reading and ran into a similar problem with the greatest lower bound:

"Suppose the set E is bounded from below. Then a lower bound zo of E is called the greatest lower bound of E if every other lower bound z of E is less than or equal to zo."

So could this be interpreted as E having an element zo being the (numerically) smallest element such that elements z of R are all smaller than this zo?

Also, what does it mean when a system of nested half open intervals have an empty intersection? Does it mean they don't share any of the same elements...?

Thanks for any replies, I appreciate any pointers.

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# Elementary Real and Complex Analysis

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