Thread: Finding the (countable) union of these events

I'm stumped on a few problems here. They are both very similar, and so I will ask just one (should be able to get the other one if I can get this one)

Suppose that for i= 1,2,3,... the set Ai is the set of real numbers from (2/i) to (3-(1/i)) inclusive. What is the countable union of these events?

What i've attempted so far:

A1 = [2,2] = { 2 }
A2 = [1, 5/2] = { x | XeR, 1<=X<=5/2}
.
..
...
Ai = [(2/i), (3-(1/i))] inslusive

So in order to find the countable union of these events, i've used the fact that as i increases, the range the set covers increases, and since each set/event contains the previous one, the union of the set will grow towards A = (0,3) (exclusive) as i tends to infinity.

I'm not sure if this is correct reasoning, or if there is a better method for attempting these problems. Also, the set A = (0,3) for real numbers is uncountably infinite.. isn't this a contradiction somewhere?

Originally Posted by mkelly09

So in order to find the countable union of these events, i've used the fact that as i increases, the range the set covers increases, and since each set/event contains the previous one, the union of the set will grow towards A = (0,3) (exclusive) as i tends to infinity.

This is correct. It's always possible to provide more details, but I am not sure it is necessary.

Originally Posted by mkelly09

Also, the set A = (0,3) for real numbers is uncountably infinite.. isn't this a contradiction somewhere?

Each for is a nonempty interval, i.e., is uncountably infinite, so it's not surprising that the union is uncountably infinite.