Let $\displaystyle \Omega = \{A_n:n= 1,2,3,...\}$ where $\displaystyle A_n = \left[\frac{1}{n}, 1 + \frac{1}{2n}\right)$. Find $\displaystyle \bigcup \Omega$
hint: graph the first element on a number line, then the second, then the third. you should of course be seeing a pattern by then.
doing it without graphing is not much harder. what happens to 1/n for various values of n? what about 1 + 1/(2n)? how do these change when n gets large?
Isn't the left endpoint getting smaller as k gets bigger?
1, 1/2, 1/3, 1/4, ... , 1/k
All intervals after $\displaystyle A_1$ fit inside of $\displaystyle A_1$... So, the arbitrary union of all the intervals would then be the one interval that contains them all... Or, that's how my professor made it seem anyway...
as i said, you seem to be having trouble visualizing this without drawing it. draw a picture. A_1 is the interval [1, 3/2), A_2 is [1/2, 5/4), notice that the second interval is outside of the scope of the first. A_2 contains the points on [1/2, 1) which are NOT in A_1, and so, A_1 cannot represent the union, because we can see that A_2 has points that are not in A_1
getting smaller means the intervals are "expanding" to the left. the left endpoint keeps getting farther from 1