What does the union and intersection of a single sequence of real number mean?

Question

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.

@David K, thank you, I think there's no detail definition of real number, it's just the second page of the book, — jimy, Jan 13 '20 at 14:28

score 1 · Answer 1 · answered Jan 13 '20 at 14:40

1

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 $$ x \vee y = \max\{x,y\}\\ x \wedge y = \min\{x,y\} $$

answered Jan 13 '20 at 14:40

GEdgar

111,679

Thank you, GEdgar, I am too careless, missed the definition in the same page. – jimy Jan 14 '20 at 13:49

What does the union and intersection of a single sequence of real number mean?

1 Answers1