SophieToft
New member
- Joined
- Oct 3, 2006
- Messages
- 17
If E is a non empty set and\(\displaystyle (B_n)_{n \geq 1}\) are elements in the set \(\displaystyle 2^E\).
I then need help showing the following:
\(\displaystyle lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcup_{n\, =\, 1} ^{\infty}\, B_n\)
if and only if \(\displaystyle B_n\, \subseteq\, B_{n+1}\), for all \(\displaystyle n\, \geq\, 1\),
Also I need to show
\(\displaystyle lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcap_{n=1} ^{\infty} B_n\)
if and only if \(\displaystyle B_n\, \supseteq\, B_{n+1}\), for all \(\displaystyle n\, \geq\, 1\)
I know that for every sequence \(\displaystyle (a_n)_{n\, \geq\, 1}\) of elements in the set \(\displaystyle - \infty\ \union\ \mathbb{R}\ \union\ \infty\).
\(\displaystyle lim_n\, sup\, a_n\, =\, inf(M_n|\, n\, \geq\, 1)\), where \(\displaystyle M_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1\).
\(\displaystyle lim_n\, inf\, a_n\, =\, inf(m_n|\, n\, \geq\, 1)\), where \(\displaystyle m_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1\).
But could somebody please give me a hint or an idear on how to use this fact to show the original task???
Sincerely Yours
Sophie Toft
_________________________________-
Edited by stapel -- Reason for edit: inserting formatting tags
I then need help showing the following:
\(\displaystyle lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcup_{n\, =\, 1} ^{\infty}\, B_n\)
if and only if \(\displaystyle B_n\, \subseteq\, B_{n+1}\), for all \(\displaystyle n\, \geq\, 1\),
Also I need to show
\(\displaystyle lim_n\, sup\, B_n\, =\, lim_n\, inf\, B_n\, =\, \bigcap_{n=1} ^{\infty} B_n\)
if and only if \(\displaystyle B_n\, \supseteq\, B_{n+1}\), for all \(\displaystyle n\, \geq\, 1\)
I know that for every sequence \(\displaystyle (a_n)_{n\, \geq\, 1}\) of elements in the set \(\displaystyle - \infty\ \union\ \mathbb{R}\ \union\ \infty\).
\(\displaystyle lim_n\, sup\, a_n\, =\, inf(M_n|\, n\, \geq\, 1)\), where \(\displaystyle M_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1\).
\(\displaystyle lim_n\, inf\, a_n\, =\, inf(m_n|\, n\, \geq\, 1)\), where \(\displaystyle m_n\, :=\, sup(a_k|\, k\, \geq\, n},\, n\, \geq\, 1\).
But could somebody please give me a hint or an idear on how to use this fact to show the original task???
Sincerely Yours
Sophie Toft
_________________________________-
Edited by stapel -- Reason for edit: inserting formatting tags
