# Thread: Convergence on the real-line

1. ## Convergence on the real-line

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. : )

2. $\displaystyle \left( {0,1} \right] = \bigcap\limits_{n \in \mathbb{Z}^ + } {\left( {0,1 + \frac{1}{n}} \right)} \;\& \,\left\{ 0 \right\} = \bigcap\limits_{n \in \mathbb{Z}^ + } {\left[ {0,\frac{1} {n}} \right)}$

3. I think I need a little bit more help answering these questions than just that. Thanks a lot though.

4. Originally Posted by AKTilted
I think I need a little bit more help answering these questions than just that. Thanks a lot though.
In that case, you need a sit-down session with a live tutor.
This is not the place to get that depth of help.

5. "Get that depth of help"? I'm just looking for more help than just one line of math with zero explanation. If you could afford that I would be much more appreciative than receiving condescending responses like in your last post...