Urgent: Measurable space theory

[SophieToft]

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
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Edited by stapel -- Reason for edit: inserting formatting tags

 

[pka]

I, for one, simply cannot read this posting.
It appears that you are using TeX.
Why not edit the posting and use the TeX properly.