Hi, I need help with the following proof question for my stats course. It states:

On the real line consider the collection of left-open/right-closed intervals:

$\displaystyle I = \left\{ (a,b] : a,b \in \mathbb{R} \right\}$

a) Write each of (0,1) and {0} explicitly as monotone limits of elements, $\displaystyle A_n$ in $\displaystyle I$.

b) For the $\displaystyle A_n$'s chosen in a), determine the lower and upper sequences

$\displaystyle B_n = \bigcap_{k \ge n }A_k , k = 1,2,...$ and

$\displaystyle C_n = \bigcap_{k \ge n}A_k, n = 1,2,...$

c) In general if $\displaystyle A_n$, n = 1,2,... is monotone prove that it does indeed converge and, explicitly, that

$\displaystyle lim A_n = \bigcup A_n$ if $\displaystyle A_n \uparrow$ and

$\displaystyle lim A_n = \bigcap A_n$ if $\displaystyle A_n \downarrow$

d) Let $\displaystyle a_n = (-1)^n/n , n = 1,2,...$ so it is clear that $\displaystyle a_n \to 0 $

For $\displaystyle A_n = (a_n,1], n = 1,2,...$ provide both its upper and lower sequences as in b) and thus prove whether or not $\displaystyle A_n$ converges.

Thanks a lot. I'm trying really hard to wrap my head around this stuff. : )