intersection of union

• Sep 12th 2009, 09:14 PM
Kat-M
intersection of union
i do not understand how this notation works. $\displaystyle \bigcap_{n=1}^\infty$$\displaystyle \bigcup_{m \geq n}$$\displaystyle A_m$
• Sep 13th 2009, 01:08 AM
Taluivren
Hi, i don't know if the following visualisation is helpful:
$\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m = \bigcap_{n=1}^\infty (A_n \cup A_{n+1}\cup A_{n+2}\cup A_{n+3}\ldots) =$

$\displaystyle = (A_1 \cup A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{2}\cup A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap (A_{3}\cup A_{4}\cup A_{5}\cup A_{6}\ldots)\cap\ldots$

but maybe this little proposition will make things clear:

$\displaystyle x \in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m \mbox{ if and only if } x \in A_k \mbox{ for infinitely many } k \in \mathbb{N}.$

Proof: "=>" assume $\displaystyle x\in A_k$ for only finitely many $\displaystyle k\in\mathbb{N}$, let $\displaystyle l$ be the biggest such $\displaystyle k$, then for all $\displaystyle m\ge l+1$ we have $\displaystyle x\not \in A_m$. This means $\displaystyle x \not \in \bigcup_{m\ge l+1}A_m$, so $\displaystyle x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$.

"<=" assume $\displaystyle x\not\in \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, this means there's some $\displaystyle n\in\mathbb{N}$ such that $\displaystyle x \not \in \bigcup_{m\ge n}A_m$, that is, $\displaystyle x\not\in A_m$ for all $\displaystyle m\ge n$, so $\displaystyle x$ belongs to only finitely many $\displaystyle A_k$.
• Sep 13th 2009, 02:10 AM
Kat-M
So if we define $\displaystyle lim_{n \rightarrow \infty}sup$ $\displaystyle A_m$=$\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, what is $\displaystyle sup$ $\displaystyle A_5$?
• Sep 13th 2009, 02:25 AM
Taluivren
Quote:

Originally Posted by Kat-M
So if we define $\displaystyle lim_{n \rightarrow \infty}sup$ $\displaystyle A_m$=$\displaystyle \bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$, what is $\displaystyle sup$ $\displaystyle A_5$?

this doesn't make sense.
it is "limes superior of a sequence of sets $\displaystyle \{A_n:\, n\in \mathbb{N}\}$" , by definition:
$\displaystyle \limsup_{n\rightarrow \infty}A_n =\bigcap_{n=1}^\infty \bigcup_{m \geq n} A_m$