Can anyone point me to two lists?
List one would be a complete list of properties, and the suggested order in which they should be proved, for sup and inf.
List two would be a complete list of properties, and the suggested order in which they should be proved, of liminf and limsup of sequences?
Thanks
David
On
Problems 2.4.12-2.4.28 in the book Wieslawa J. Kaczor, Maria T. Nowak: Problems in mathematical analysis: Volume 1; Real Numbers, Sequences and Series, give several useful properties of $\liminf$ and $\limsup$. Since the text is intended as book with exercise to practice, they are organized in the order such that what you need for previous exercises is mentioned before you get to them.
This series of problems starts on p.42. The book also contains solutions. If I remember correctly, the original was written in Polish, there exist French and English translations of the book.
Of course such a complete list does not exist. However, a very useful short and nearly complete list does exist. By nearly complete I mean that it gives a few things you need to know about $\sup$ and $\inf$ that will enable you to easily argue about anything related (no guarantees made here!).
So, first, I'm assuming here $\sup$ and $\inf$ are of non-empty sets of real numbers. Rule number 1: $\sup$ and $\inf$ are dual. Whatever holds for one, holds for the other as long as you remember to reverse the order. Rule number 2: Know the definition: Given a set $S\subseteq \mathbb R$ and $a\in \mathbb R$, it holds that $a=\sup S$ precisely when $a$ is an upper bound of $S$ and whenever $t$ is an upper bound of $S$, then $s\le t$ (this is what least upper bound means). Here is now your first opportunity to practice rule 1: give the definition of $\inf$. Rule number 3: $\sup S\le x$ iff $s\le x$ holds for all $s\in S$. Rule number 4: if $y\le s$ for some $s\in S$, then $y\le \sup S$. That's it.