I read measure_theory by Paul R-Halmos, part of number 7 prerequisite concept of reading this book is:
"The supremum and infimum of a sequence {$x_n$} of real numbers are denoted by
$\bigcup_{i =1}^\infty x_i$ and $\bigcap_{i =1}^\infty x_i$."
Why would a union or intersection of a real number sequence exist? For example, number 2 and 9, what does '$2\bigcup9$' mean?
I only understand union and intersection between sets, thanks for paying attention for my question.
Just before that on the same page, Halmos defines $$ x \cup y = \max\{x,y\}\\ x \cap y = \min\{x,y\} $$ That will explain what you want to know. There is need to mention Dedekind cuts.
Note: This notation, perhaps common in 1950, is no longer common. Nowadays we might define instead $$ x \vee y = \max\{x,y\}\\ x \wedge y = \min\{x,y\} $$