# Homework Help: Probability - Infinite Union of Subsets of a Sample Space

1. Apr 10, 2010

### Legendre

1. The problem statement, all variables and given/known data

This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach.

Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t}
where Y is a random variable and t is any real number.

Fix a real number X.

Consider an increasing sequence {Xn} such that Xn tend to X from below/left as n tend to infinity. Therefore, Xn < or = X for all n.

So, A(Xn) is a subset of A(X) for all n.

2. Relevant equations

N.A.

3. The attempt at a solution

This isn't a homework question but I thought it fits in this forum. It is stated in my text that the infinite union of A(Xn) is not A(X).

I think this is because Xn can tend to X in such a way that X is not equal to X all for n. So their infinite union will always not contain some element in X.

But how do I show this mathematically? A little nudge in the right direction so I can solve this myself? Thanks in advance!